Examples

$S = {(1, 1, 1), (0, 1, 1)}$

Step 1

Assign a name for each vector.

${u⃗}_{1} = (1, 1, 1)$

${u⃗}_{2} = (0, 1, 1)$

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 orthonormal basis by dividing each orthogonal vector by its norm.

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

Step 6

Find the unit vector $\frac{{v⃗}_{1}}{| | {v⃗}_{1} | |}$ where ${v⃗}_{1} = (1, 1, 1)$ .

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Step 6.1

To find a unit vector in the same direction as a vector $v⃗$ , divide by the norm of $v⃗$ .

$\frac{v⃗}{| v⃗ |}$

Step 6.2

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 6.3

Simplify.

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Step 6.3.1

One to any power is one.

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

Step 6.3.2

One to any power is one.

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

Step 6.3.3

One to any power is one.

$\sqrt{1 + 1 + 1}$

Step 6.3.4

Add $1$ and $1$ .

$\sqrt{2 + 1}$

Step 6.3.5

Add $2$ and $1$ .

$\sqrt{3}$

Step 6.4

Divide the vector by its norm.

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

Step 6.5

Divide each element in the vector by $\sqrt{3}$ .

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

Step 7

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 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.

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

Step 7.3

Simplify.

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Step 7.3.1

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

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Step 7.3.1.1

Apply the product rule to $- \frac{2}{3}$ .

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

Step 7.3.1.2

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 7.3.2

Raise $- 1$ to the power of $2$ .

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

Step 7.3.3

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

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

Step 7.3.4

Raise $2$ to the power of $2$ .

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

Step 7.3.5

Raise $3$ to the power of $2$ .

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

Step 7.3.6

Apply the product rule to $\frac{1}{3}$ .

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

Step 7.3.7

One to any power is one.

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

Step 7.3.8

Raise $3$ to the power of $2$ .

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

Step 7.3.9

Apply the product rule to $\frac{1}{3}$ .

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

Step 7.3.10

One to any power is one.

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

Step 7.3.11

Raise $3$ to the power of $2$ .

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

Step 7.3.12

Combine the numerators over the common denominator.

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

Step 7.3.13

Add $4$ and $1$ .

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

Step 7.3.14

Combine the numerators over the common denominator.

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

Step 7.3.15

Add $5$ and $1$ .

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

Step 7.3.16

Cancel the common factor of $6$ and $9$ .

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Step 7.3.16.1

Factor $3$ out of $6$ .

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

Step 7.3.16.2

Cancel the common factors.

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Step 7.3.16.2.1

Factor $3$ out of $9$ .

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

Step 7.3.16.2.2

Cancel the common factor.

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

Step 7.3.16.2.3

Rewrite the expression.

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

Step 7.3.17

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

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

Step 7.4

Divide the vector by its norm.

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

Step 7.5

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 7.6

Simplify.

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Step 7.6.1

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 7.6.2

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 7.6.3

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 7.6.4

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 7.6.5

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 7.6.6

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 7.6.7

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 7.6.8

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 8

Substitute the known values.

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

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