Linear Algebra Examples

$(1, - 1, 2)$ , $(0, 3, 1)$

Step 1

Use the cross product formula to find the angle between two vectors.

$θ = \arcsin (\frac{| a⃗ \times b⃗ |}{| a⃗ | | b⃗ |})$

Step 2

Find the cross product.

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Step 2.1

The cross product of two vectors $a⃗$ and $b⃗$ can be written as a determinant with the standard unit vectors from $ℝ^{3}$ and the elements of the given vectors.

$a⃗ \times b⃗ = a⃗ \times b⃗ = | \begin{array}{c} î & ĵ & k̂ \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{array} |$

Step 2.2

Set up the determinant with the given values.

$a⃗ \times b⃗ = | \begin{array}{c} î & ĵ & k̂ \\ 1 & - 1 & 2 \\ 0 & 3 & 1 \end{array} |$

Step 2.3

Choose the row or column with the most $0$ elements. If there are no $0$ elements choose any row or column. Multiply every element in row $1$ by its cofactor and add.

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Step 2.3.1

Consider the corresponding sign chart.

$| \begin{array}{c} + & - & + \\ - & + & - \\ + & - & + \end{array} |$

Step 2.3.2

The cofactor is the minor with the sign changed if the indices match a $-$ position on the sign chart.

Step 2.3.3

The minor for $a_{11}$ is the determinant with row $1$ and column $1$ deleted.

$| \begin{array}{c} - 1 & 2 \\ 3 & 1 \end{array} |$

Step 2.3.4

Multiply element $a_{11}$ by its cofactor.

$| \begin{array}{c} - 1 & 2 \\ 3 & 1 \end{array} | î$

Step 2.3.5

The minor for $a_{12}$ is the determinant with row $1$ and column $2$ deleted.

$| \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} |$

Step 2.3.6

Multiply element $a_{12}$ by its cofactor.

$- | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | ĵ$

Step 2.3.7

The minor for $a_{13}$ is the determinant with row $1$ and column $3$ deleted.

$| \begin{array}{c} 1 & - 1 \\ 0 & 3 \end{array} |$

Step 2.3.8

Multiply element $a_{13}$ by its cofactor.

$| \begin{array}{c} 1 & - 1 \\ 0 & 3 \end{array} | k̂$

Step 2.3.9

Add the terms together.

$a⃗ \times b⃗ = | \begin{array}{c} - 1 & 2 \\ 3 & 1 \end{array} | î - | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | ĵ + | \begin{array}{c} 1 & - 1 \\ 0 & 3 \end{array} | k̂$

Step 2.4

Evaluate $| \begin{array}{c} - 1 & 2 \\ 3 & 1 \end{array} |$ .

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Step 2.4.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a⃗ \times b⃗ = (- 1 \cdot 1 - 3 \cdot 2) î - | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | ĵ + | \begin{array}{c} 1 & - 1 \\ 0 & 3 \end{array} | k̂$

Step 2.4.2

Simplify the determinant.

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Step 2.4.2.1

Simplify each term.

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Step 2.4.2.1.1

Multiply $- 1$ by $1$ .

$a⃗ \times b⃗ = (- 1 - 3 \cdot 2) î - | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | ĵ + | \begin{array}{c} 1 & - 1 \\ 0 & 3 \end{array} | k̂$

Step 2.4.2.1.2

Multiply $- 3$ by $2$ .

$a⃗ \times b⃗ = (- 1 - 6) î - | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | ĵ + | \begin{array}{c} 1 & - 1 \\ 0 & 3 \end{array} | k̂$

Step 2.4.2.2

Subtract $6$ from $- 1$ .

$a⃗ \times b⃗ = - 7 î - | \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} | ĵ + | \begin{array}{c} 1 & - 1 \\ 0 & 3 \end{array} | k̂$

Step 2.5

Evaluate $| \begin{array}{c} 1 & 2 \\ 0 & 1 \end{array} |$ .

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Step 2.5.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a⃗ \times b⃗ = - 7 î - (1 \cdot 1 + 0 \cdot 2) ĵ + | \begin{array}{c} 1 & - 1 \\ 0 & 3 \end{array} | k̂$

Step 2.5.2

Simplify the determinant.

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Step 2.5.2.1

Simplify each term.

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Step 2.5.2.1.1

Multiply $1$ by $1$ .

$a⃗ \times b⃗ = - 7 î - (1 + 0 \cdot 2) ĵ + | \begin{array}{c} 1 & - 1 \\ 0 & 3 \end{array} | k̂$

Step 2.5.2.1.2

Multiply $0$ by $2$ .

$a⃗ \times b⃗ = - 7 î - (1 + 0) ĵ + | \begin{array}{c} 1 & - 1 \\ 0 & 3 \end{array} | k̂$

Step 2.5.2.2

Add $1$ and $0$ .

$a⃗ \times b⃗ = - 7 î - 1 \cdot 1 ĵ + | \begin{array}{c} 1 & - 1 \\ 0 & 3 \end{array} | k̂$

Step 2.6

Evaluate $| \begin{array}{c} 1 & - 1 \\ 0 & 3 \end{array} |$ .

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Step 2.6.1

The determinant of a $2 \times 2$ matrix can be found using the formula $| \begin{array}{c} a & b \\ c & d \end{array} | = a d - c b$ .

$a⃗ \times b⃗ = - 7 î - 1 \cdot 1 ĵ + (1 \cdot 3 + 0 \cdot - 1) k̂$

Step 2.6.2

Simplify the determinant.

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Step 2.6.2.1

Simplify each term.

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Step 2.6.2.1.1

Multiply $3$ by $1$ .

$a⃗ \times b⃗ = - 7 î - 1 \cdot 1 ĵ + (3 + 0 \cdot - 1) k̂$

Step 2.6.2.1.2

Multiply $0$ by $- 1$ .

$a⃗ \times b⃗ = - 7 î - 1 \cdot 1 ĵ + (3 + 0) k̂$

Step 2.6.2.2

Add $3$ and $0$ .

$a⃗ \times b⃗ = - 7 î - 1 \cdot 1 ĵ + 3 k̂$

Step 2.7

Multiply $- 1$ by $1$ .

$a⃗ \times b⃗ = - 7 î - ĵ + 3 k̂$

Step 2.8

Rewrite the answer.

$a⃗ \times b⃗ = (- 7, - 1, 3)$

Step 3

Find the magnitude of the cross product.

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Step 3.1

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

$| a⃗ \times b⃗ | = \sqrt{{(- 7)}^{2} + {(- 1)}^{2} + 3^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

Raise $- 7$ to the power of $2$ .

$| a⃗ \times b⃗ | = \sqrt{49 + {(- 1)}^{2} + 3^{2}}$

Step 3.2.2

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

$| a⃗ \times b⃗ | = \sqrt{49 + 1 + 3^{2}}$

Step 3.2.3

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

$| a⃗ \times b⃗ | = \sqrt{49 + 1 + 9}$

Step 3.2.4

Add $49$ and $1$ .

$| a⃗ \times b⃗ | = \sqrt{50 + 9}$

Step 3.2.5

Add $50$ and $9$ .

$| a⃗ \times b⃗ | = \sqrt{59}$

Step 4

Find the magnitude of $a⃗$ .

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Step 4.1

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

$| a⃗ | = \sqrt{1^{2} + {(- 1)}^{2} + 2^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

One to any power is one.

$| a⃗ | = \sqrt{1 + {(- 1)}^{2} + 2^{2}}$

Step 4.2.2

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

$| a⃗ | = \sqrt{1 + 1 + 2^{2}}$

Step 4.2.3

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

$| a⃗ | = \sqrt{1 + 1 + 4}$

Step 4.2.4

Add $1$ and $1$ .

$| a⃗ | = \sqrt{2 + 4}$

Step 4.2.5

Add $2$ and $4$ .

$| a⃗ | = \sqrt{6}$

Step 5

Find the magnitude of $b⃗$ .

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Step 5.1

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

$| b⃗ | = \sqrt{0^{2} + 3^{2} + 1^{2}}$

Step 5.2

Simplify.

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Step 5.2.1

Raising $0$ to any positive power yields $0$ .

$| b⃗ | = \sqrt{0 + 3^{2} + 1^{2}}$

Step 5.2.2

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

$| b⃗ | = \sqrt{0 + 9 + 1^{2}}$

Step 5.2.3

One to any power is one.

$| b⃗ | = \sqrt{0 + 9 + 1}$

Step 5.2.4

Add $0$ and $9$ .

$| b⃗ | = \sqrt{9 + 1}$

Step 5.2.5

Add $9$ and $1$ .

$| b⃗ | = \sqrt{10}$

Step 6

Substitute the values into the formula.

$θ = \arcsin (\frac{\sqrt{59}}{\sqrt{6} \sqrt{10}})$

Step 7

Simplify.

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Step 7.1

Simplify the denominator.

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Step 7.1.1

Combine using the product rule for radicals.

$θ = \arcsin (\frac{\sqrt{59}}{\sqrt{6 \cdot 10}})$

Step 7.1.2

Multiply $6$ by $10$ .

$θ = \arcsin (\frac{\sqrt{59}}{\sqrt{60}})$

Step 7.2

Simplify the denominator.

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Step 7.2.1

Rewrite $60$ as $2^{2} \cdot 15$ .

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Step 7.2.1.1

Factor $4$ out of $60$ .

$θ = \arcsin (\frac{\sqrt{59}}{\sqrt{4 (15)}})$

Step 7.2.1.2

Rewrite $4$ as $2^{2}$ .

$θ = \arcsin (\frac{\sqrt{59}}{\sqrt{2^{2} \cdot 15}})$

Step 7.2.2

Pull terms out from under the radical.

$θ = \arcsin (\frac{\sqrt{59}}{2 \sqrt{15}})$

Step 7.3

Multiply $\frac{\sqrt{59}}{2 \sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$θ = \arcsin (\frac{\sqrt{59}}{2 \sqrt{15}} \cdot \frac{\sqrt{15}}{\sqrt{15}})$

Step 7.4

Combine and simplify the denominator.

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Step 7.4.1

Multiply $\frac{\sqrt{59}}{2 \sqrt{15}}$ by $\frac{\sqrt{15}}{\sqrt{15}}$ .

$θ = \arcsin (\frac{\sqrt{59} \sqrt{15}}{2 \sqrt{15} \sqrt{15}})$

Step 7.4.2

Move $\sqrt{15}$ .

$θ = \arcsin (\frac{\sqrt{59} \sqrt{15}}{2 (\sqrt{15} \sqrt{15})})$

Step 7.4.3

Raise $\sqrt{15}$ to the power of $1$ .

$θ = \arcsin (\frac{\sqrt{59} \sqrt{15}}{2 ({\sqrt{15}}^{1} \sqrt{15})})$

Step 7.4.4

Raise $\sqrt{15}$ to the power of $1$ .

$θ = \arcsin (\frac{\sqrt{59} \sqrt{15}}{2 ({\sqrt{15}}^{1} {\sqrt{15}}^{1})})$

Step 7.4.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$θ = \arcsin (\frac{\sqrt{59} \sqrt{15}}{2 {\sqrt{15}}^{1 + 1}})$

Step 7.4.6

Add $1$ and $1$ .

$θ = \arcsin (\frac{\sqrt{59} \sqrt{15}}{2 {\sqrt{15}}^{2}})$

Step 7.4.7

Rewrite ${\sqrt{15}}^{2}$ as $15$ .

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Step 7.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{15}$ as $15^{\frac{1}{2}}$ .

$θ = \arcsin (\frac{\sqrt{59} \sqrt{15}}{2 {(15^{\frac{1}{2}})}^{2}})$

Step 7.4.7.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$θ = \arcsin (\frac{\sqrt{59} \sqrt{15}}{2 \cdot 15^{\frac{1}{2} \cdot 2}})$

Step 7.4.7.3

Combine $\frac{1}{2}$ and $2$ .

$θ = \arcsin (\frac{\sqrt{59} \sqrt{15}}{2 \cdot 15^{\frac{2}{2}}})$

Step 7.4.7.4

Cancel the common factor of $2$ .

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Step 7.4.7.4.1

Cancel the common factor.

$θ = \arcsin (\frac{\sqrt{59} \sqrt{15}}{2 \cdot 15^{\frac{2}{2}}})$

Step 7.4.7.4.2

Rewrite the expression.

$θ = \arcsin (\frac{\sqrt{59} \sqrt{15}}{2 \cdot 15^{1}})$

Step 7.4.7.5

Evaluate the exponent.

$θ = \arcsin (\frac{\sqrt{59} \sqrt{15}}{2 \cdot 15})$

Step 7.5

Simplify the numerator.

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Step 7.5.1

Combine using the product rule for radicals.

$θ = \arcsin (\frac{\sqrt{59 \cdot 15}}{2 \cdot 15})$

Step 7.5.2

Multiply $59$ by $15$ .

$θ = \arcsin (\frac{\sqrt{885}}{2 \cdot 15})$

Step 7.6

Multiply $2$ by $15$ .

$θ = \arcsin (\frac{\sqrt{885}}{30})$

Step 7.7

Evaluate $\arcsin (\frac{\sqrt{885}}{30})$ .

$θ = 82.5824442$

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