Trigonometry Examples

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

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

Step 1

Assign a name for each vector.

{u⃗}_{1} = (1, 1, 1)

${u⃗}_{1} = (1, 1, 1)$

{u⃗}_{2} = (0, 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)

${v⃗}_{1} = {u⃗}_{1} = (1, 1, 1)$

Step 3

Use the formula to find the other orthogonal vectors.

{v⃗}_{k} = {u⃗}_{k} - k - 1 \sum i = 1 {proj}_{{v⃗}_{i}} ({u⃗}_{k})

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

Step 4

Find the orthogonal vector

{v⃗}_{2}

${v⃗}_{2}$ .

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Step 4.1

Use the formula to find

{v⃗}_{2}

${v⃗}_{2}$ .

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

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

Step 4.2

Substitute

(0, 1, 1)

$(0, 1, 1)$ for

{u⃗}_{2}

${u⃗}_{2}$ .

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

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

Step 4.3

Find

{proj}_{{v⃗}_{1}} ({u⃗}_{2})

${proj}_{{v⃗}_{1}} ({u⃗}_{2})$ .

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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

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

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

$0$ by

1

$1$ .

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

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

Step 4.3.1.2.1.2

Multiply

1

$1$ by

1

$1$ .

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

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

Step 4.3.1.2.1.3

Multiply

1

$1$ by

1

$1$ .

{u⃗}_{2} \cdot {v⃗}_{1} = 0 + 1 + 1

${u⃗}_{2} \cdot {v⃗}_{1} = 0 + 1 + 1$

{u⃗}_{2} \cdot {v⃗}_{1} = 0 + 1 + 1

${u⃗}_{2} \cdot {v⃗}_{1} = 0 + 1 + 1$

Step 4.3.1.2.2

Add

0

$0$ and

1

$1$ .

{u⃗}_{2} \cdot {v⃗}_{1} = 1 + 1

${u⃗}_{2} \cdot {v⃗}_{1} = 1 + 1$

Step 4.3.1.2.3

Add

1

$1$ and

1

$1$ .

{u⃗}_{2} \cdot {v⃗}_{1} = 2

${u⃗}_{2} \cdot {v⃗}_{1} = 2$

{u⃗}_{2} \cdot {v⃗}_{1} = 2

${u⃗}_{2} \cdot {v⃗}_{1} = 2$

{u⃗}_{2} \cdot {v⃗}_{1} = 2

${u⃗}_{2} \cdot {v⃗}_{1} = 2$

Step 4.3.2

Find the norm of

{v⃗}_{1} = (1, 1, 1)

${v⃗}_{1} = (1, 1, 1)$ .

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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}}

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

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}}

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

Step 4.3.2.2.2

One to any power is one.

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

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

Step 4.3.2.2.3

One to any power is one.

| | {v⃗}_{1} | | = \sqrt{1 + 1 + 1}

$| | {v⃗}_{1} | | = \sqrt{1 + 1 + 1}$

Step 4.3.2.2.4

Add

1

$1$ and

1

$1$ .

| | {v⃗}_{1} | | = \sqrt{2 + 1}

$| | {v⃗}_{1} | | = \sqrt{2 + 1}$

Step 4.3.2.2.5

Add

2

$2$ and

1

$1$ .

| | {v⃗}_{1} | | = \sqrt{3}

$| | {v⃗}_{1} | | = \sqrt{3}$

| | {v⃗}_{1} | | = \sqrt{3}

$| | {v⃗}_{1} | | = \sqrt{3}$

| | {v⃗}_{1} | | = \sqrt{3}

$| | {v⃗}_{1} | | = \sqrt{3}$

Step 4.3.3

Find the projection of

{u⃗}_{2}

${u⃗}_{2}$ onto

{v⃗}_{1}

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

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

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

Step 4.3.4

Substitute

2

$2$ for

{u⃗}_{2} \cdot {v⃗}_{1}

${u⃗}_{2} \cdot {v⃗}_{1}$ .

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

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

Step 4.3.5

Substitute

\sqrt{3}

$\sqrt{3}$ for

| | {v⃗}_{1} | |

$| | {v⃗}_{1} | |$ .

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

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

Step 4.3.6

Substitute

(1, 1, 1)

$(1, 1, 1)$ for

{v⃗}_{1}

${v⃗}_{1}$ .

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

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

3

$3$ .

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Step 4.3.7.1.1

Use

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

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

\sqrt{3}

$\sqrt{3}$ as

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

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

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

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

Step 4.3.7.1.2

Apply the power rule and multiply exponents,

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

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

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

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

Step 4.3.7.1.3

Combine

\frac{1}{2}

$\frac{1}{2}$ and

2

$2$ .

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

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

Step 4.3.7.1.4

Cancel the common factor of

2

$2$ .

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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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