Examples

(1, 1, 1)

$(1, 1, 1)$ ,

$(0, 1, 1)$ ,

$(0, 0, 1)$

Step 1

Assign a name for each vector.

${u⃗}_{1} = (1, 1, 1)$

${u⃗}_{2} = (0, 1, 1)$

${u⃗}_{3} = (0, 0, 1)$

Step 2

The first orthogonal vector is the first vector in the given set of vectors.

${v⃗}_{1} = {u⃗}_{1} = (1, 1, 1)$

Step 3

Use the formula to find the other orthogonal vectors.

${v⃗}_{k} = {u⃗}_{k} - \sum_{i = 1}^{k - 1} {proj}_{{v⃗}_{i}} ({u⃗}_{k})$

Step 4

Find the orthogonal vector

${v⃗}_{2}$ .

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Step 4.1

Use the formula to find

${v⃗}_{2}$ .

${v⃗}_{2} = {u⃗}_{2} - {proj}_{{v⃗}_{1}} ({u⃗}_{2})$

Step 4.2

Substitute

$(0, 1, 1)$ for

${u⃗}_{2}$ .

${v⃗}_{2} = (0, 1, 1) - {proj}_{{v⃗}_{1}} ({u⃗}_{2})$

Step 4.3

Find

${proj}_{{v⃗}_{1}} ({u⃗}_{2})$ .

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Step 4.3.1

Find the dot product.

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Step 4.3.1.1

The dot product of two vectors is the sum of the products of the their components.

${u⃗}_{2} \cdot {v⃗}_{1} = 0 \cdot 1 + 1 \cdot 1 + 1 \cdot 1$

Step 4.3.1.2

Simplify.

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Step 4.3.1.2.1

Simplify each term.

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Step 4.3.1.2.1.1

Multiply

$0$ by

$1$ .

${u⃗}_{2} \cdot {v⃗}_{1} = 0 + 1 \cdot 1 + 1 \cdot 1$

Step 4.3.1.2.1.2

Multiply

$1$ by

$1$ .

${u⃗}_{2} \cdot {v⃗}_{1} = 0 + 1 + 1 \cdot 1$

Step 4.3.1.2.1.3

Multiply

$1$ by

$1$ .

${u⃗}_{2} \cdot {v⃗}_{1} = 0 + 1 + 1$

Step 4.3.1.2.2

Add

$0$ and

$1$ .

${u⃗}_{2} \cdot {v⃗}_{1} = 1 + 1$

Step 4.3.1.2.3

Add

$1$ and

$1$ .

${u⃗}_{2} \cdot {v⃗}_{1} = 2$

Step 4.3.2

Find the norm of

${v⃗}_{1} = (1, 1, 1)$ .

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Step 4.3.2.1

The norm is the square root of the sum of squares of each element in the vector.

$| | {v⃗}_{1} | | = \sqrt{1^{2} + 1^{2} + 1^{2}}$

Step 4.3.2.2

Simplify.

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Step 4.3.2.2.1

One to any power is one.

$| | {v⃗}_{1} | | = \sqrt{1 + 1^{2} + 1^{2}}$

Step 4.3.2.2.2

One to any power is one.

$| | {v⃗}_{1} | | = \sqrt{1 + 1 + 1^{2}}$

Step 4.3.2.2.3

One to any power is one.

$| | {v⃗}_{1} | | = \sqrt{1 + 1 + 1}$

Step 4.3.2.2.4

Add

$1$ and

$1$ .

$| | {v⃗}_{1} | | = \sqrt{2 + 1}$

Step 4.3.2.2.5

Add

$2$ and

$1$ .

$| | {v⃗}_{1} | | = \sqrt{3}$

Step 4.3.3

Find the projection of

${u⃗}_{2}$ onto

${v⃗}_{1}$ using the projection formula.

${proj}_{{v⃗}_{1}} ({u⃗}_{2}) = \frac{{u⃗}_{2} \cdot {v⃗}_{1}}{{| | {v⃗}_{1} | |}^{2}} \times {v⃗}_{1}$

Step 4.3.4

Substitute

$2$ for

${u⃗}_{2} \cdot {v⃗}_{1}$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{2}) = \frac{2}{{| | {v⃗}_{1} | |}^{2}} \times {v⃗}_{1}$

Step 4.3.5

Substitute

$\sqrt{3}$ for

$| | {v⃗}_{1} | |$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{2}) = \frac{2}{{\sqrt{3}}^{2}} \times {v⃗}_{1}$

Step 4.3.6

Substitute

$(1, 1, 1)$ for

${v⃗}_{1}$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{2}) = \frac{2}{{\sqrt{3}}^{2}} \times (1, 1, 1)$

Step 4.3.7

Simplify the right side.

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Step 4.3.7.1

Rewrite

${\sqrt{3}}^{2}$ as

$3$ .

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Step 4.3.7.1.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{3}$ as

$3^{\frac{1}{2}}$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{2}) = \frac{2}{{(3^{\frac{1}{2}})}^{2}} \times (1, 1, 1)$

Step 4.3.7.1.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{2}) = \frac{2}{3^{\frac{1}{2} \cdot 2}} \times (1, 1, 1)$

Step 4.3.7.1.3

Combine

$\frac{1}{2}$ and

$2$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{2}) = \frac{2}{3^{\frac{2}{2}}} \times (1, 1, 1)$

Step 4.3.7.1.4

Cancel the common factor of

$2$ .

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Step 4.3.7.1.4.1

Cancel the common factor.

${proj}_{{v⃗}_{1}} ({u⃗}_{2}) = \frac{2}{3^{\frac{2}{2}}} \times (1, 1, 1)$

Step 4.3.7.1.4.2

Rewrite the expression.

${proj}_{{v⃗}_{1}} ({u⃗}_{2}) = \frac{2}{3^{1}} \times (1, 1, 1)$

Step 4.3.7.1.5

Evaluate the exponent.

${proj}_{{v⃗}_{1}} ({u⃗}_{2}) = \frac{2}{3} \times (1, 1, 1)$

Step 4.3.7.2

Multiply

$\frac{2}{3}$ by each element of the matrix.

${proj}_{{v⃗}_{1}} ({u⃗}_{2}) = (\frac{2}{3} \cdot 1, \frac{2}{3} \cdot 1, \frac{2}{3} \cdot 1)$

Step 4.3.7.3

Simplify each element in the matrix.

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Step 4.3.7.3.1

Multiply

$\frac{2}{3}$ by

$1$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{2}) = (\frac{2}{3}, \frac{2}{3} \cdot 1, \frac{2}{3} \cdot 1)$

Step 4.3.7.3.2

Multiply

$\frac{2}{3}$ by

$1$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{2}) = (\frac{2}{3}, \frac{2}{3}, \frac{2}{3} \cdot 1)$

Step 4.3.7.3.3

Multiply

$\frac{2}{3}$ by

$1$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{2}) = (\frac{2}{3}, \frac{2}{3}, \frac{2}{3})$

Step 4.4

Substitute the projection.

${v⃗}_{2} = (0, 1, 1) - (\frac{2}{3}, \frac{2}{3}, \frac{2}{3})$

Step 4.5

Simplify.

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Step 4.5.1

Combine each component of the vectors.

$(0 - (\frac{2}{3}), 1 - (\frac{2}{3}), 1 - (\frac{2}{3}))$

Step 4.5.2

Subtract

$\frac{2}{3}$ from

$0$ .

$(- \frac{2}{3}, 1 - (\frac{2}{3}), 1 - (\frac{2}{3}))$

Step 4.5.3

Write

$1$ as a fraction with a common denominator.

$(- \frac{2}{3}, \frac{3}{3} - \frac{2}{3}, 1 - (\frac{2}{3}))$

Step 4.5.4

Combine the numerators over the common denominator.

$(- \frac{2}{3}, \frac{3 - 2}{3}, 1 - (\frac{2}{3}))$

Step 4.5.5

Subtract

$2$ from

$3$ .

$(- \frac{2}{3}, \frac{1}{3}, 1 - (\frac{2}{3}))$

Step 4.5.6

Write

$1$ as a fraction with a common denominator.

$(- \frac{2}{3}, \frac{1}{3}, \frac{3}{3} - \frac{2}{3})$

Step 4.5.7

Combine the numerators over the common denominator.

$(- \frac{2}{3}, \frac{1}{3}, \frac{3 - 2}{3})$

Step 4.5.8

Subtract

$2$ from

$3$ .

${v⃗}_{2} = (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5

Find the orthogonal vector

${v⃗}_{3}$ .

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Step 5.1

Use the formula to find

${v⃗}_{3}$ .

${v⃗}_{3} = {u⃗}_{3} - {proj}_{{v⃗}_{1}} ({u⃗}_{3}) - {proj}_{{v⃗}_{2}} ({u⃗}_{3})$

Step 5.2

Substitute

$(0, 0, 1)$ for

${u⃗}_{3}$ .

${v⃗}_{3} = (0, 0, 1) - {proj}_{{v⃗}_{1}} ({u⃗}_{3}) - {proj}_{{v⃗}_{2}} ({u⃗}_{3})$

Step 5.3

Find

${proj}_{{v⃗}_{1}} ({u⃗}_{3})$ .

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Step 5.3.1

Find the dot product.

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Step 5.3.1.1

The dot product of two vectors is the sum of the products of the their components.

${u⃗}_{3} \cdot {v⃗}_{1} = 0 \cdot 1 + 0 \cdot 1 + 1 \cdot 1$

Step 5.3.1.2

Simplify.

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Step 5.3.1.2.1

Simplify each term.

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Step 5.3.1.2.1.1

Multiply

$0$ by

$1$ .

${u⃗}_{3} \cdot {v⃗}_{1} = 0 + 0 \cdot 1 + 1 \cdot 1$

Step 5.3.1.2.1.2

Multiply

$0$ by

$1$ .

${u⃗}_{3} \cdot {v⃗}_{1} = 0 + 0 + 1 \cdot 1$

Step 5.3.1.2.1.3

Multiply

$1$ by

$1$ .

${u⃗}_{3} \cdot {v⃗}_{1} = 0 + 0 + 1$

Step 5.3.1.2.2

Add

$0$ and

$0$ .

${u⃗}_{3} \cdot {v⃗}_{1} = 0 + 1$

Step 5.3.1.2.3

Add

$0$ and

$1$ .

${u⃗}_{3} \cdot {v⃗}_{1} = 1$

Step 5.3.2

Find the norm of

${v⃗}_{1} = (1, 1, 1)$ .

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Step 5.3.2.1

The norm is the square root of the sum of squares of each element in the vector.

$| | {v⃗}_{1} | | = \sqrt{1^{2} + 1^{2} + 1^{2}}$

Step 5.3.2.2

Simplify.

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Step 5.3.2.2.1

One to any power is one.

$| | {v⃗}_{1} | | = \sqrt{1 + 1^{2} + 1^{2}}$

Step 5.3.2.2.2

One to any power is one.

$| | {v⃗}_{1} | | = \sqrt{1 + 1 + 1^{2}}$

Step 5.3.2.2.3

One to any power is one.

$| | {v⃗}_{1} | | = \sqrt{1 + 1 + 1}$

Step 5.3.2.2.4

Add

$1$ and

$1$ .

$| | {v⃗}_{1} | | = \sqrt{2 + 1}$

Step 5.3.2.2.5

Add

$2$ and

$1$ .

$| | {v⃗}_{1} | | = \sqrt{3}$

Step 5.3.3

Find the projection of

${u⃗}_{3}$ onto

${v⃗}_{1}$ using the projection formula.

${proj}_{{v⃗}_{1}} ({u⃗}_{3}) = \frac{{u⃗}_{3} \cdot {v⃗}_{1}}{{| | {v⃗}_{1} | |}^{2}} \times {v⃗}_{1}$

Step 5.3.4

Substitute

$1$ for

${u⃗}_{3} \cdot {v⃗}_{1}$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{3}) = \frac{1}{{| | {v⃗}_{1} | |}^{2}} \times {v⃗}_{1}$

Step 5.3.5

Substitute

$\sqrt{3}$ for

$| | {v⃗}_{1} | |$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{3}) = \frac{1}{{\sqrt{3}}^{2}} \times {v⃗}_{1}$

Step 5.3.6

Substitute

$(1, 1, 1)$ for

${v⃗}_{1}$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{3}) = \frac{1}{{\sqrt{3}}^{2}} \times (1, 1, 1)$

Step 5.3.7

Simplify the right side.

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Step 5.3.7.1

Rewrite

${\sqrt{3}}^{2}$ as

$3$ .

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Step 5.3.7.1.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{3}$ as

$3^{\frac{1}{2}}$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{3}) = \frac{1}{{(3^{\frac{1}{2}})}^{2}} \times (1, 1, 1)$

Step 5.3.7.1.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{3}) = \frac{1}{3^{\frac{1}{2} \cdot 2}} \times (1, 1, 1)$

Step 5.3.7.1.3

Combine

$\frac{1}{2}$ and

$2$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{3}) = \frac{1}{3^{\frac{2}{2}}} \times (1, 1, 1)$

Step 5.3.7.1.4

Cancel the common factor of

$2$ .

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Step 5.3.7.1.4.1

Cancel the common factor.

${proj}_{{v⃗}_{1}} ({u⃗}_{3}) = \frac{1}{3^{\frac{2}{2}}} \times (1, 1, 1)$

Step 5.3.7.1.4.2

Rewrite the expression.

${proj}_{{v⃗}_{1}} ({u⃗}_{3}) = \frac{1}{3^{1}} \times (1, 1, 1)$

Step 5.3.7.1.5

Evaluate the exponent.

${proj}_{{v⃗}_{1}} ({u⃗}_{3}) = \frac{1}{3} \times (1, 1, 1)$

Step 5.3.7.2

Multiply

$\frac{1}{3}$ by each element of the matrix.

${proj}_{{v⃗}_{1}} ({u⃗}_{3}) = (\frac{1}{3} \cdot 1, \frac{1}{3} \cdot 1, \frac{1}{3} \cdot 1)$

Step 5.3.7.3

Simplify each element in the matrix.

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Step 5.3.7.3.1

Multiply

$\frac{1}{3}$ by

$1$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{3}) = (\frac{1}{3}, \frac{1}{3} \cdot 1, \frac{1}{3} \cdot 1)$

Step 5.3.7.3.2

Multiply

$\frac{1}{3}$ by

$1$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{3}) = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3} \cdot 1)$

Step 5.3.7.3.3

Multiply

$\frac{1}{3}$ by

$1$ .

${proj}_{{v⃗}_{1}} ({u⃗}_{3}) = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4

Find

${proj}_{{v⃗}_{2}} ({u⃗}_{3})$ .

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Step 5.4.1

Find the dot product.

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Step 5.4.1.1

The dot product of two vectors is the sum of the products of the their components.

${u⃗}_{3} \cdot {v⃗}_{2} = 0 (- \frac{2}{3}) + 0 (\frac{1}{3}) + 1 (\frac{1}{3})$

Step 5.4.1.2

Simplify.

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Step 5.4.1.2.1

Simplify each term.

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Step 5.4.1.2.1.1

Multiply

$0 (- \frac{2}{3})$ .

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Step 5.4.1.2.1.1.1

Multiply

$- 1$ by

$0$ .

${u⃗}_{3} \cdot {v⃗}_{2} = 0 (\frac{2}{3}) + 0 (\frac{1}{3}) + 1 (\frac{1}{3})$

Step 5.4.1.2.1.1.2

Multiply

$0$ by

$\frac{2}{3}$ .

${u⃗}_{3} \cdot {v⃗}_{2} = 0 + 0 (\frac{1}{3}) + 1 (\frac{1}{3})$

Step 5.4.1.2.1.2

Multiply

$0$ by

$\frac{1}{3}$ .

${u⃗}_{3} \cdot {v⃗}_{2} = 0 + 0 + 1 (\frac{1}{3})$

Step 5.4.1.2.1.3

Multiply

$\frac{1}{3}$ by

$1$ .

${u⃗}_{3} \cdot {v⃗}_{2} = 0 + 0 + \frac{1}{3}$

Step 5.4.1.2.2

Add

$0$ and

$0$ .

${u⃗}_{3} \cdot {v⃗}_{2} = 0 + \frac{1}{3}$

Step 5.4.1.2.3

Add

$0$ and

$\frac{1}{3}$ .

${u⃗}_{3} \cdot {v⃗}_{2} = \frac{1}{3}$

Step 5.4.2

Find the norm of

${v⃗}_{2} = (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$ .

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Step 5.4.2.1

The norm is the square root of the sum of squares of each element in the vector.

$| | {v⃗}_{2} | | = \sqrt{{(- \frac{2}{3})}^{2} + {(\frac{1}{3})}^{2} + {(\frac{1}{3})}^{2}}$

Step 5.4.2.2

Simplify.

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Step 5.4.2.2.1

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 5.4.2.2.1.1

Apply the product rule to

$- \frac{2}{3}$ .

$| | {v⃗}_{2} | | = \sqrt{{(- 1)}^{2} {(\frac{2}{3})}^{2} + {(\frac{1}{3})}^{2} + {(\frac{1}{3})}^{2}}$

Step 5.4.2.2.1.2

Apply the product rule to

$\frac{2}{3}$ .

$| | {v⃗}_{2} | | = \sqrt{{(- 1)}^{2} \frac{2^{2}}{3^{2}} + {(\frac{1}{3})}^{2} + {(\frac{1}{3})}^{2}}$

Step 5.4.2.2.2

Raise

$- 1$ to the power of

$2$ .

$| | {v⃗}_{2} | | = \sqrt{1 \frac{2^{2}}{3^{2}} + {(\frac{1}{3})}^{2} + {(\frac{1}{3})}^{2}}$

Step 5.4.2.2.3

Multiply

$\frac{2^{2}}{3^{2}}$ by

$1$ .

$| | {v⃗}_{2} | | = \sqrt{\frac{2^{2}}{3^{2}} + {(\frac{1}{3})}^{2} + {(\frac{1}{3})}^{2}}$

Step 5.4.2.2.4

Raise

$2$ to the power of

$2$ .

$| | {v⃗}_{2} | | = \sqrt{\frac{4}{3^{2}} + {(\frac{1}{3})}^{2} + {(\frac{1}{3})}^{2}}$

Step 5.4.2.2.5

Raise

$3$ to the power of

$2$ .

$| | {v⃗}_{2} | | = \sqrt{\frac{4}{9} + {(\frac{1}{3})}^{2} + {(\frac{1}{3})}^{2}}$

Step 5.4.2.2.6

Apply the product rule to

$\frac{1}{3}$ .

$| | {v⃗}_{2} | | = \sqrt{\frac{4}{9} + \frac{1^{2}}{3^{2}} + {(\frac{1}{3})}^{2}}$

Step 5.4.2.2.7

One to any power is one.

$| | {v⃗}_{2} | | = \sqrt{\frac{4}{9} + \frac{1}{3^{2}} + {(\frac{1}{3})}^{2}}$

Step 5.4.2.2.8

Raise

$3$ to the power of

$2$ .

$| | {v⃗}_{2} | | = \sqrt{\frac{4}{9} + \frac{1}{9} + {(\frac{1}{3})}^{2}}$

Step 5.4.2.2.9

Apply the product rule to

$\frac{1}{3}$ .

$| | {v⃗}_{2} | | = \sqrt{\frac{4}{9} + \frac{1}{9} + \frac{1^{2}}{3^{2}}}$

Step 5.4.2.2.10

One to any power is one.

$| | {v⃗}_{2} | | = \sqrt{\frac{4}{9} + \frac{1}{9} + \frac{1}{3^{2}}}$

Step 5.4.2.2.11

Raise

$3$ to the power of

$2$ .

$| | {v⃗}_{2} | | = \sqrt{\frac{4}{9} + \frac{1}{9} + \frac{1}{9}}$

Step 5.4.2.2.12

Combine the numerators over the common denominator.

$| | {v⃗}_{2} | | = \sqrt{\frac{4 + 1}{9} + \frac{1}{9}}$

Step 5.4.2.2.13

Add

$4$ and

$1$ .

$| | {v⃗}_{2} | | = \sqrt{\frac{5}{9} + \frac{1}{9}}$

Step 5.4.2.2.14

Combine the numerators over the common denominator.

$| | {v⃗}_{2} | | = \sqrt{\frac{5 + 1}{9}}$

Step 5.4.2.2.15

Add

$5$ and

$1$ .

$| | {v⃗}_{2} | | = \sqrt{\frac{6}{9}}$

Step 5.4.2.2.16

Cancel the common factor of

$6$ and

$9$ .

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Step 5.4.2.2.16.1

Factor

$3$ out of

$6$ .

$| | {v⃗}_{2} | | = \sqrt{\frac{3 (2)}{9}}$

Step 5.4.2.2.16.2

Cancel the common factors.

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Step 5.4.2.2.16.2.1

Factor

$3$ out of

$9$ .

$| | {v⃗}_{2} | | = \sqrt{\frac{3 \cdot 2}{3 \cdot 3}}$

Step 5.4.2.2.16.2.2

Cancel the common factor.

$| | {v⃗}_{2} | | = \sqrt{\frac{3 \cdot 2}{3 \cdot 3}}$

Step 5.4.2.2.16.2.3

Rewrite the expression.

$| | {v⃗}_{2} | | = \sqrt{\frac{2}{3}}$

Step 5.4.2.2.17

Rewrite

$\sqrt{\frac{2}{3}}$ as

$\frac{\sqrt{2}}{\sqrt{3}}$ .

$| | {v⃗}_{2} | | = \frac{\sqrt{2}}{\sqrt{3}}$

Step 5.4.2.2.18

Multiply

$\frac{\sqrt{2}}{\sqrt{3}}$ by

$\frac{\sqrt{3}}{\sqrt{3}}$ .

$| | {v⃗}_{2} | | = \frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}}$

Step 5.4.2.2.19

Combine and simplify the denominator.

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Step 5.4.2.2.19.1

Multiply

$\frac{\sqrt{2}}{\sqrt{3}}$ by

$\frac{\sqrt{3}}{\sqrt{3}}$ .

$| | {v⃗}_{2} | | = \frac{\sqrt{2} \sqrt{3}}{\sqrt{3} \sqrt{3}}$

Step 5.4.2.2.19.2

Raise

$\sqrt{3}$ to the power of

$1$ .

$| | {v⃗}_{2} | | = \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} \sqrt{3}}$

Step 5.4.2.2.19.3

Raise

$\sqrt{3}$ to the power of

$1$ .

$| | {v⃗}_{2} | | = \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1} {\sqrt{3}}^{1}}$

Step 5.4.2.2.19.4

Use the power rule

$a^{m} a^{n} = a^{m + n}$ to combine exponents.

$| | {v⃗}_{2} | | = \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{1 + 1}}$

Step 5.4.2.2.19.5

Add

$1$ and

$1$ .

$| | {v⃗}_{2} | | = \frac{\sqrt{2} \sqrt{3}}{{\sqrt{3}}^{2}}$

Step 5.4.2.2.19.6

Rewrite

${\sqrt{3}}^{2}$ as

$3$ .

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Step 5.4.2.2.19.6.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{3}$ as

$3^{\frac{1}{2}}$ .

$| | {v⃗}_{2} | | = \frac{\sqrt{2} \sqrt{3}}{{(3^{\frac{1}{2}})}^{2}}$

Step 5.4.2.2.19.6.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

$| | {v⃗}_{2} | | = \frac{\sqrt{2} \sqrt{3}}{3^{\frac{1}{2} \cdot 2}}$

Step 5.4.2.2.19.6.3

Combine

$\frac{1}{2}$ and

$2$ .

$| | {v⃗}_{2} | | = \frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.4.2.2.19.6.4

Cancel the common factor of

$2$ .

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Step 5.4.2.2.19.6.4.1

Cancel the common factor.

$| | {v⃗}_{2} | | = \frac{\sqrt{2} \sqrt{3}}{3^{\frac{2}{2}}}$

Step 5.4.2.2.19.6.4.2

Rewrite the expression.

$| | {v⃗}_{2} | | = \frac{\sqrt{2} \sqrt{3}}{3^{1}}$

Step 5.4.2.2.19.6.5

Evaluate the exponent.

$| | {v⃗}_{2} | | = \frac{\sqrt{2} \sqrt{3}}{3}$

Step 5.4.2.2.20

Simplify the numerator.

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Step 5.4.2.2.20.1

Combine using the product rule for radicals.

$| | {v⃗}_{2} | | = \frac{\sqrt{2 \cdot 3}}{3}$

Step 5.4.2.2.20.2

Multiply

$2$ by

$3$ .

$| | {v⃗}_{2} | | = \frac{\sqrt{6}}{3}$

Step 5.4.3

Find the projection of

${u⃗}_{3}$ onto

${v⃗}_{2}$ using the projection formula.

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{{u⃗}_{3} \cdot {v⃗}_{2}}{{| | {v⃗}_{2} | |}^{2}} \times {v⃗}_{2}$

Step 5.4.4

Substitute

$\frac{1}{3}$ for

${u⃗}_{3} \cdot {v⃗}_{2}$ .

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{\frac{1}{3}}{{| | {v⃗}_{2} | |}^{2}} \times {v⃗}_{2}$

Step 5.4.5

Substitute

$\frac{\sqrt{6}}{3}$ for

$| | {v⃗}_{2} | |$ .

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{\frac{1}{3}}{{(\frac{\sqrt{6}}{3})}^{2}} \times {v⃗}_{2}$

Step 5.4.6

Substitute

$(- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$ for

${v⃗}_{2}$ .

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{\frac{1}{3}}{{(\frac{\sqrt{6}}{3})}^{2}} \times (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4.7

Simplify the right side.

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Step 5.4.7.1

Simplify the denominator.

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Step 5.4.7.1.1

Apply the product rule to

$\frac{\sqrt{6}}{3}$ .

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{\frac{1}{3}}{\frac{{\sqrt{6}}^{2}}{3^{2}}} \times (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4.7.1.2

Rewrite

${\sqrt{6}}^{2}$ as

$6$ .

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Step 5.4.7.1.2.1

Use

$\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite

$\sqrt{6}$ as

$6^{\frac{1}{2}}$ .

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{\frac{1}{3}}{\frac{{(6^{\frac{1}{2}})}^{2}}{3^{2}}} \times (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4.7.1.2.2

Apply the power rule and multiply exponents,

${(a^{m})}^{n} = a^{m n}$ .

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{\frac{1}{3}}{\frac{6^{\frac{1}{2} \cdot 2}}{3^{2}}} \times (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4.7.1.2.3

Combine

$\frac{1}{2}$ and

$2$ .

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{\frac{1}{3}}{\frac{6^{\frac{2}{2}}}{3^{2}}} \times (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4.7.1.2.4

Cancel the common factor of

$2$ .

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Step 5.4.7.1.2.4.1

Cancel the common factor.

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{\frac{1}{3}}{\frac{6^{\frac{2}{2}}}{3^{2}}} \times (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4.7.1.2.4.2

Rewrite the expression.

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{\frac{1}{3}}{\frac{6^{1}}{3^{2}}} \times (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4.7.1.2.5

Evaluate the exponent.

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{\frac{1}{3}}{\frac{6}{3^{2}}} \times (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4.7.1.3

Raise

$3$ to the power of

$2$ .

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{\frac{1}{3}}{\frac{6}{9}} \times (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4.7.1.4

Cancel the common factor of

$6$ and

$9$ .

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Step 5.4.7.1.4.1

Factor

$3$ out of

$6$ .

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{\frac{1}{3}}{\frac{3 (2)}{9}} \times (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4.7.1.4.2

Cancel the common factors.

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Step 5.4.7.1.4.2.1

Factor

$3$ out of

$9$ .

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{\frac{1}{3}}{\frac{3 \cdot 2}{3 \cdot 3}} \times (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4.7.1.4.2.2

Cancel the common factor.

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{\frac{1}{3}}{\frac{3 \cdot 2}{3 \cdot 3}} \times (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4.7.1.4.2.3

Rewrite the expression.

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{\frac{1}{3}}{\frac{2}{3}} \times (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4.7.2

Multiply the numerator by the reciprocal of the denominator.

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{1}{3} \cdot \frac{3}{2} \times (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4.7.3

Cancel the common factor of

$3$ .

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Step 5.4.7.3.1

Cancel the common factor.

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{1}{3} \cdot \frac{3}{2} \times (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4.7.3.2

Rewrite the expression.

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = \frac{1}{2} \times (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$

Step 5.4.7.4

Multiply

$\frac{1}{2}$ by each element of the matrix.

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = (\frac{1}{2} (- \frac{2}{3}), \frac{1}{2} \cdot \frac{1}{3}, \frac{1}{2} \cdot \frac{1}{3})$

Step 5.4.7.5

Simplify each element in the matrix.

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Step 5.4.7.5.1

Cancel the common factor of

$2$ .

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Step 5.4.7.5.1.1

Move the leading negative in

$- \frac{2}{3}$ into the numerator.

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = (\frac{1}{2} \cdot \frac{- 2}{3}, \frac{1}{2} \cdot \frac{1}{3}, \frac{1}{2} \cdot \frac{1}{3})$

Step 5.4.7.5.1.2

Factor

$2$ out of

$- 2$ .

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = (\frac{1}{2} \cdot \frac{2 (- 1)}{3}, \frac{1}{2} \cdot \frac{1}{3}, \frac{1}{2} \cdot \frac{1}{3})$

Step 5.4.7.5.1.3

Cancel the common factor.

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = (\frac{1}{2} \cdot \frac{2 \cdot - 1}{3}, \frac{1}{2} \cdot \frac{1}{3}, \frac{1}{2} \cdot \frac{1}{3})$

Step 5.4.7.5.1.4

Rewrite the expression.

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = (\frac{- 1}{3}, \frac{1}{2} \cdot \frac{1}{3}, \frac{1}{2} \cdot \frac{1}{3})$

Step 5.4.7.5.2

Move the negative in front of the fraction.

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = (- \frac{1}{3}, \frac{1}{2} \cdot \frac{1}{3}, \frac{1}{2} \cdot \frac{1}{3})$

Step 5.4.7.5.3

Multiply

$\frac{1}{2} \cdot \frac{1}{3}$ .

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Step 5.4.7.5.3.1

Multiply

$\frac{1}{2}$ by

$\frac{1}{3}$ .

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = (- \frac{1}{3}, \frac{1}{2 \cdot 3}, \frac{1}{2} \cdot \frac{1}{3})$

Step 5.4.7.5.3.2

Multiply

$2$ by

$3$ .

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = (- \frac{1}{3}, \frac{1}{6}, \frac{1}{2} \cdot \frac{1}{3})$

Step 5.4.7.5.4

Multiply

$\frac{1}{2} \cdot \frac{1}{3}$ .

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Step 5.4.7.5.4.1

Multiply

$\frac{1}{2}$ by

$\frac{1}{3}$ .

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = (- \frac{1}{3}, \frac{1}{6}, \frac{1}{2 \cdot 3})$

Step 5.4.7.5.4.2

Multiply

$2$ by

$3$ .

${proj}_{{v⃗}_{2}} ({u⃗}_{3}) = (- \frac{1}{3}, \frac{1}{6}, \frac{1}{6})$

Step 5.5

Substitute the projections.

${v⃗}_{3} = (0, 0, 1) - (\frac{1}{3}, \frac{1}{3}, \frac{1}{3}) - (- \frac{1}{3}, \frac{1}{6}, \frac{1}{6})$

Step 5.6

Simplify.

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Step 5.6.1

Combine each component of the vectors.

$(0 - (\frac{1}{3}), 0 - (\frac{1}{3}), 1 - (\frac{1}{3})) - (- \frac{1}{3}, \frac{1}{6}, \frac{1}{6})$

Step 5.6.2

Combine each component of the vectors.

$(0 - (\frac{1}{3}) - (- \frac{1}{3}), 0 - (\frac{1}{3}) - (\frac{1}{6}), 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.3

Multiply

$- (- \frac{1}{3})$ .

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Step 5.6.3.1

Multiply

$- 1$ by

$- 1$ .

$(0 - \frac{1}{3} + 1 (\frac{1}{3}), 0 - (\frac{1}{3}) - (\frac{1}{6}), 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.3.2

Multiply

$\frac{1}{3}$ by

$1$ .

$(0 - \frac{1}{3} + \frac{1}{3}, 0 - (\frac{1}{3}) - (\frac{1}{6}), 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.4

Combine fractions.

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Step 5.6.4.1

Combine the numerators over the common denominator.

$(\frac{- 1 + 1}{3}, 0 - (\frac{1}{3}) - (\frac{1}{6}), 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.4.2

Simplify the expression.

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Step 5.6.4.2.1

Add

$- 1$ and

$1$ .

$(\frac{0}{3}, 0 - (\frac{1}{3}) - (\frac{1}{6}), 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.4.2.2

Divide

$0$ by

$3$ .

$(0, 0 - (\frac{1}{3}) - (\frac{1}{6}), 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.5

Multiply

$- 1$ by

$\frac{1}{6}$ .

$(0, 0 - \frac{1}{3} - \frac{1}{6}, 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.6

Subtract

$\frac{1}{3}$ from

$0$ .

$(0, - \frac{1}{3} - \frac{1}{6}, 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.7

To write

$- \frac{1}{3}$ as a fraction with a common denominator, multiply by

$\frac{2}{2}$ .

$(0, - \frac{1}{3} \cdot \frac{2}{2} - \frac{1}{6}, 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.8

Write each expression with a common denominator of

$6$ , by multiplying each by an appropriate factor of

$1$ .

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Step 5.6.8.1

Multiply

$\frac{1}{3}$ by

$\frac{2}{2}$ .

$(0, - \frac{2}{3 \cdot 2} - \frac{1}{6}, 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.8.2

Multiply

$3$ by

$2$ .

$(0, - \frac{2}{6} - \frac{1}{6}, 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.9

Simplify the expression.

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Step 5.6.9.1

Combine the numerators over the common denominator.

$(0, \frac{- 2 - 1}{6}, 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.9.2

Subtract

$1$ from

$- 2$ .

$(0, \frac{- 3}{6}, 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.10

Cancel the common factor of

$- 3$ and

$6$ .

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Step 5.6.10.1

Factor

$3$ out of

$- 3$ .

$(0, \frac{3 (- 1)}{6}, 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.10.2

Cancel the common factors.

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Step 5.6.10.2.1

Factor

$3$ out of

$6$ .

$(0, \frac{3 \cdot - 1}{3 \cdot 2}, 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.10.2.2

Cancel the common factor.

$(0, \frac{3 \cdot - 1}{3 \cdot 2}, 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.10.2.3

Rewrite the expression.

$(0, \frac{- 1}{2}, 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.11

Move the negative in front of the fraction.

$(0, - \frac{1}{2}, 1 - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.12

Find the common denominator.

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Step 5.6.12.1

Write

$1$ as a fraction with denominator

$1$ .

$(0, - \frac{1}{2}, \frac{1}{1} - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.12.2

Multiply

$\frac{1}{1}$ by

$\frac{6}{6}$ .

$(0, - \frac{1}{2}, \frac{1}{1} \cdot \frac{6}{6} - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.12.3

Multiply

$\frac{1}{1}$ by

$\frac{6}{6}$ .

$(0, - \frac{1}{2}, \frac{6}{6} - (\frac{1}{3}) - (\frac{1}{6}))$

Step 5.6.12.4

Multiply

$\frac{1}{3}$ by

$\frac{2}{2}$ .

$(0, - \frac{1}{2}, \frac{6}{6} - (\frac{1}{3} \cdot \frac{2}{2}) - (\frac{1}{6}))$

Step 5.6.12.5

Multiply

$\frac{1}{3}$ by

$\frac{2}{2}$ .

$(0, - \frac{1}{2}, \frac{6}{6} - \frac{2}{3 \cdot 2} - (\frac{1}{6}))$

Step 5.6.12.6

Reorder the factors of

$3 \cdot 2$ .

$(0, - \frac{1}{2}, \frac{6}{6} - \frac{2}{2 \cdot 3} - (\frac{1}{6}))$

Step 5.6.12.7

Multiply

$2$ by

$3$ .

$(0, - \frac{1}{2}, \frac{6}{6} - \frac{2}{6} - (\frac{1}{6}))$

Step 5.6.13

Combine the numerators over the common denominator.

$(0, - \frac{1}{2}, \frac{6 - 2 - 1}{6})$

Step 5.6.14

Simplify by subtracting numbers.

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Step 5.6.14.1

Subtract

$2$ from

$6$ .

$(0, - \frac{1}{2}, \frac{4 - 1}{6})$

Step 5.6.14.2

Subtract

$1$ from

$4$ .

$(0, - \frac{1}{2}, \frac{3}{6})$

Step 5.6.15

Cancel the common factor of

$3$ and

$6$ .

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Step 5.6.15.1

Factor

$3$ out of

$3$ .

$(0, - \frac{1}{2}, \frac{3 (1)}{6})$

Step 5.6.15.2

Cancel the common factors.

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Step 5.6.15.2.1

Factor

$3$ out of

$6$ .

$(0, - \frac{1}{2}, \frac{3 \cdot 1}{3 \cdot 2})$

Step 5.6.15.2.2

Cancel the common factor.

$(0, - \frac{1}{2}, \frac{3 \cdot 1}{3 \cdot 2})$

Step 5.6.15.2.3

Rewrite the expression.

${v⃗}_{3} = (0, - \frac{1}{2}, \frac{1}{2})$

Step 6

Find the orthonormal basis by dividing each orthogonal vector by its norm.

$Span {\frac{{v⃗}_{1}}{| | {v⃗}_{1} | |}, \frac{{v⃗}_{2}}{| | {v⃗}_{2} | |}, \frac{{v⃗}_{3}}{| | {v⃗}_{3} | |}}$

Step 7

Find the unit vector

$\frac{{v⃗}_{1}}{| | {v⃗}_{1} | |}$ where

${v⃗}_{1} = (1, 1, 1)$ .

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Step 7.1

To find a unit vector in the same direction as a vector

$v⃗$ , divide by the norm of

$v⃗$ .

$\frac{v⃗}{| v⃗ |}$

Step 7.2

The norm is the square root of the sum of squares of each element in the vector.

$\sqrt{1^{2} + 1^{2} + 1^{2}}$

Step 7.3

Simplify.

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Step 7.3.1

One to any power is one.

$\sqrt{1 + 1^{2} + 1^{2}}$

Step 7.3.2

One to any power is one.

$\sqrt{1 + 1 + 1^{2}}$

Step 7.3.3

One to any power is one.

$\sqrt{1 + 1 + 1}$

Step 7.3.4

Add

$1$ and

$1$ .

$\sqrt{2 + 1}$

Step 7.3.5

Add

$2$ and

$1$ .

$\sqrt{3}$

Step 7.4

Divide the vector by its norm.

$\frac{(1, 1, 1)}{\sqrt{3}}$

Step 7.5

Divide each element in the vector by

$\sqrt{3}$ .

$(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}})$

Step 8

Find the unit vector

$\frac{{v⃗}_{2}}{| | {v⃗}_{2} | |}$ where

${v⃗}_{2} = (- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})$ .

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Step 8.1

To find a unit vector in the same direction as a vector

$v⃗$ , divide by the norm of

$v⃗$ .

$\frac{v⃗}{| v⃗ |}$

Step 8.2

The norm is the square root of the sum of squares of each element in the vector.

$\sqrt{{(- \frac{2}{3})}^{2} + {(\frac{1}{3})}^{2} + {(\frac{1}{3})}^{2}}$

Step 8.3

Simplify.

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Step 8.3.1

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 8.3.1.1

Apply the product rule to

$- \frac{2}{3}$ .

$\sqrt{{(- 1)}^{2} {(\frac{2}{3})}^{2} + {(\frac{1}{3})}^{2} + {(\frac{1}{3})}^{2}}$

Step 8.3.1.2

Apply the product rule to

$\frac{2}{3}$ .

$\sqrt{{(- 1)}^{2} \frac{2^{2}}{3^{2}} + {(\frac{1}{3})}^{2} + {(\frac{1}{3})}^{2}}$

Step 8.3.2

Raise

$- 1$ to the power of

$2$ .

$\sqrt{1 \frac{2^{2}}{3^{2}} + {(\frac{1}{3})}^{2} + {(\frac{1}{3})}^{2}}$

Step 8.3.3

Multiply

$\frac{2^{2}}{3^{2}}$ by

$1$ .

$\sqrt{\frac{2^{2}}{3^{2}} + {(\frac{1}{3})}^{2} + {(\frac{1}{3})}^{2}}$

Step 8.3.4

Raise

$2$ to the power of

$2$ .

$\sqrt{\frac{4}{3^{2}} + {(\frac{1}{3})}^{2} + {(\frac{1}{3})}^{2}}$

Step 8.3.5

Raise

$3$ to the power of

$2$ .

$\sqrt{\frac{4}{9} + {(\frac{1}{3})}^{2} + {(\frac{1}{3})}^{2}}$

Step 8.3.6

Apply the product rule to

$\frac{1}{3}$ .

$\sqrt{\frac{4}{9} + \frac{1^{2}}{3^{2}} + {(\frac{1}{3})}^{2}}$

Step 8.3.7

One to any power is one.

$\sqrt{\frac{4}{9} + \frac{1}{3^{2}} + {(\frac{1}{3})}^{2}}$

Step 8.3.8

Raise

$3$ to the power of

$2$ .

$\sqrt{\frac{4}{9} + \frac{1}{9} + {(\frac{1}{3})}^{2}}$

Step 8.3.9

Apply the product rule to

$\frac{1}{3}$ .

$\sqrt{\frac{4}{9} + \frac{1}{9} + \frac{1^{2}}{3^{2}}}$

Step 8.3.10

One to any power is one.

$\sqrt{\frac{4}{9} + \frac{1}{9} + \frac{1}{3^{2}}}$

Step 8.3.11

Raise

$3$ to the power of

$2$ .

$\sqrt{\frac{4}{9} + \frac{1}{9} + \frac{1}{9}}$

Step 8.3.12

Combine the numerators over the common denominator.

$\sqrt{\frac{4 + 1}{9} + \frac{1}{9}}$

Step 8.3.13

Add

$4$ and

$1$ .

$\sqrt{\frac{5}{9} + \frac{1}{9}}$

Step 8.3.14

Combine the numerators over the common denominator.

$\sqrt{\frac{5 + 1}{9}}$

Step 8.3.15

Add

$5$ and

$1$ .

$\sqrt{\frac{6}{9}}$

Step 8.3.16

Cancel the common factor of

$6$ and

$9$ .

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Step 8.3.16.1

Factor

$3$ out of

$6$ .

$\sqrt{\frac{3 (2)}{9}}$

Step 8.3.16.2

Cancel the common factors.

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Step 8.3.16.2.1

Factor

$3$ out of

$9$ .

$\sqrt{\frac{3 \cdot 2}{3 \cdot 3}}$

Step 8.3.16.2.2

Cancel the common factor.

$\sqrt{\frac{3 \cdot 2}{3 \cdot 3}}$

Step 8.3.16.2.3

Rewrite the expression.

$\sqrt{\frac{2}{3}}$

Step 8.3.17

Rewrite

$\sqrt{\frac{2}{3}}$ as

$\frac{\sqrt{2}}{\sqrt{3}}$ .

$\frac{\sqrt{2}}{\sqrt{3}}$

Step 8.4

Divide the vector by its norm.

$\frac{(- \frac{2}{3}, \frac{1}{3}, \frac{1}{3})}{\frac{\sqrt{2}}{\sqrt{3}}}$

Step 8.5

Divide each element in the vector by

$\frac{\sqrt{2}}{\sqrt{3}}$ .

$(\frac{- \frac{2}{3}}{\frac{\sqrt{2}}{\sqrt{3}}}, \frac{\frac{1}{3}}{\frac{\sqrt{2}}{\sqrt{3}}}, \frac{\frac{1}{3}}{\frac{\sqrt{2}}{\sqrt{3}}})$

Step 8.6

Simplify.

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Step 8.6.1

Multiply the numerator by the reciprocal of the denominator.

$(- \frac{2}{3} \cdot \frac{\sqrt{3}}{\sqrt{2}}, \frac{\frac{1}{3}}{\frac{\sqrt{2}}{\sqrt{3}}}, \frac{\frac{1}{3}}{\frac{\sqrt{2}}{\sqrt{3}}})$

Step 8.6.2

Multiply

$\frac{\sqrt{3}}{\sqrt{2}}$ by

$\frac{2}{3}$ .

$(- \frac{\sqrt{3} \cdot 2}{\sqrt{2} \cdot 3}, \frac{\frac{1}{3}}{\frac{\sqrt{2}}{\sqrt{3}}}, \frac{\frac{1}{3}}{\frac{\sqrt{2}}{\sqrt{3}}})$

Step 8.6.3

Move

$2$ to the left of

$\sqrt{3}$ .

$(- \frac{2 \sqrt{3}}{\sqrt{2} \cdot 3}, \frac{\frac{1}{3}}{\frac{\sqrt{2}}{\sqrt{3}}}, \frac{\frac{1}{3}}{\frac{\sqrt{2}}{\sqrt{3}}})$

Step 8.6.4

Move

$3$ to the left of

$\sqrt{2}$ .

$(- \frac{2 \sqrt{3}}{3 \sqrt{2}}, \frac{\frac{1}{3}}{\frac{\sqrt{2}}{\sqrt{3}}}, \frac{\frac{1}{3}}{\frac{\sqrt{2}}{\sqrt{3}}})$

Step 8.6.5

Multiply the numerator by the reciprocal of the denominator.

$(- \frac{2 \sqrt{3}}{3 \sqrt{2}}, \frac{1}{3} \cdot \frac{\sqrt{3}}{\sqrt{2}}, \frac{\frac{1}{3}}{\frac{\sqrt{2}}{\sqrt{3}}})$

Step 8.6.6

Multiply

$\frac{1}{3}$ by

$\frac{\sqrt{3}}{\sqrt{2}}$ .

$(- \frac{2 \sqrt{3}}{3 \sqrt{2}}, \frac{\sqrt{3}}{3 \sqrt{2}}, \frac{\frac{1}{3}}{\frac{\sqrt{2}}{\sqrt{3}}})$

Step 8.6.7

Multiply the numerator by the reciprocal of the denominator.

$(- \frac{2 \sqrt{3}}{3 \sqrt{2}}, \frac{\sqrt{3}}{3 \sqrt{2}}, \frac{1}{3} \cdot \frac{\sqrt{3}}{\sqrt{2}})$

Step 8.6.8

Multiply

$\frac{1}{3}$ by

$\frac{\sqrt{3}}{\sqrt{2}}$ .

$(- \frac{2 \sqrt{3}}{3 \sqrt{2}}, \frac{\sqrt{3}}{3 \sqrt{2}}, \frac{\sqrt{3}}{3 \sqrt{2}})$

Step 9

Find the unit vector

$\frac{{v⃗}_{3}}{| | {v⃗}_{3} | |}$ where

${v⃗}_{3} = (0, - \frac{1}{2}, \frac{1}{2})$ .

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Step 9.1

To find a unit vector in the same direction as a vector

$v⃗$ , divide by the norm of

$v⃗$ .

$\frac{v⃗}{| v⃗ |}$

Step 9.2

The norm is the square root of the sum of squares of each element in the vector.

$\sqrt{0^{2} + {(- \frac{1}{2})}^{2} + {(\frac{1}{2})}^{2}}$

Step 9.3

Simplify.

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Step 9.3.1

Raising

$0$ to any positive power yields

$0$ .

$\sqrt{0 + {(- \frac{1}{2})}^{2} + {(\frac{1}{2})}^{2}}$

Step 9.3.2

Use the power rule

${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

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Step 9.3.2.1

Apply the product rule to

$- \frac{1}{2}$ .

$\sqrt{0 + {(- 1)}^{2} {(\frac{1}{2})}^{2} + {(\frac{1}{2})}^{2}}$

Step 9.3.2.2

Apply the product rule to

$\frac{1}{2}$ .

$\sqrt{0 + {(- 1)}^{2} \frac{1^{2}}{2^{2}} + {(\frac{1}{2})}^{2}}$

Step 9.3.3

Raise

$- 1$ to the power of

$2$ .

$\sqrt{0 + 1 \frac{1^{2}}{2^{2}} + {(\frac{1}{2})}^{2}}$

Step 9.3.4

Multiply

$\frac{1^{2}}{2^{2}}$ by

$1$ .

$\sqrt{0 + \frac{1^{2}}{2^{2}} + {(\frac{1}{2})}^{2}}$

Step 9.3.5

One to any power is one.

$\sqrt{0 + \frac{1}{2^{2}} + {(\frac{1}{2})}^{2}}$

Step 9.3.6

Raise

$2$ to the power of

$2$ .

$\sqrt{0 + \frac{1}{4} + {(\frac{1}{2})}^{2}}$

Step 9.3.7

Apply the product rule to

$\frac{1}{2}$ .

$\sqrt{0 + \frac{1}{4} + \frac{1^{2}}{2^{2}}}$

Step 9.3.8

One to any power is one.

$\sqrt{0 + \frac{1}{4} + \frac{1}{2^{2}}}$

Step 9.3.9

Raise

$2$ to the power of

$2$ .

$\sqrt{0 + \frac{1}{4} + \frac{1}{4}}$

Step 9.3.10

Add

$0$ and

$\frac{1}{4}$ .

$\sqrt{\frac{1}{4} + \frac{1}{4}}$

Step 9.3.11

Combine the numerators over the common denominator.

$\sqrt{\frac{1 + 1}{4}}$

Step 9.3.12

Add

$1$ and

$1$ .

$\sqrt{\frac{2}{4}}$

Step 9.3.13

Cancel the common factor of

$2$ and

$4$ .

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Step 9.3.13.1

Factor

$2$ out of

$2$ .

$\sqrt{\frac{2 (1)}{4}}$

Step 9.3.13.2

Cancel the common factors.

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Step 9.3.13.2.1

Factor

$2$ out of

$4$ .

$\sqrt{\frac{2 \cdot 1}{2 \cdot 2}}$

Step 9.3.13.2.2

Cancel the common factor.

$\sqrt{\frac{2 \cdot 1}{2 \cdot 2}}$

Step 9.3.13.2.3

Rewrite the expression.

$\sqrt{\frac{1}{2}}$

Step 9.3.14

Rewrite

$\sqrt{\frac{1}{2}}$ as

$\frac{\sqrt{1}}{\sqrt{2}}$ .

$\frac{\sqrt{1}}{\sqrt{2}}$

Step 9.3.15

Any root of

$1$ is

$1$ .

$\frac{1}{\sqrt{2}}$

Step 9.4

Divide the vector by its norm.

$\frac{(0, - \frac{1}{2}, \frac{1}{2})}{\frac{1}{\sqrt{2}}}$

Step 9.5

Divide each element in the vector by

$\frac{1}{\sqrt{2}}$ .

$(\frac{0}{\frac{1}{\sqrt{2}}}, \frac{- \frac{1}{2}}{\frac{1}{\sqrt{2}}}, \frac{\frac{1}{2}}{\frac{1}{\sqrt{2}}})$

Step 9.6

Simplify.

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Step 9.6.1

Multiply the numerator by the reciprocal of the denominator.

$(0 \sqrt{2}, \frac{- \frac{1}{2}}{\frac{1}{\sqrt{2}}}, \frac{\frac{1}{2}}{\frac{1}{\sqrt{2}}})$

Step 9.6.2

Multiply

$0$ by

$\sqrt{2}$ .

$(0, \frac{- \frac{1}{2}}{\frac{1}{\sqrt{2}}}, \frac{\frac{1}{2}}{\frac{1}{\sqrt{2}}})$

Step 9.6.3

Multiply the numerator by the reciprocal of the denominator.

$(0, - \frac{1}{2} \sqrt{2}, \frac{\frac{1}{2}}{\frac{1}{\sqrt{2}}})$

Step 9.6.4

Combine

$\sqrt{2}$ and

$\frac{1}{2}$ .

$(0, - \frac{\sqrt{2}}{2}, \frac{\frac{1}{2}}{\frac{1}{\sqrt{2}}})$

Step 9.6.5

Multiply the numerator by the reciprocal of the denominator.

$(0, - \frac{\sqrt{2}}{2}, \frac{1}{2} \sqrt{2})$

Step 9.6.6

Combine

$\frac{1}{2}$ and

$\sqrt{2}$ .

$(0, - \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

Step 10

Substitute the known values.

$Span {(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}), (- \frac{2 \sqrt{3}}{3 \sqrt{2}}, \frac{\sqrt{3}}{3 \sqrt{2}}, \frac{\sqrt{3}}{3 \sqrt{2}}), (0, - \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})}$

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