Linear Algebra Examples

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

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.

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

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.

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

Step 2.3.4

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

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

Step 2.3.5

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

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

Step 2.3.6

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

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

Step 2.3.7

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

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

Step 2.3.8

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

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

Step 2.3.9

Add the terms together.

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

Step 2.4

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

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Step 2.4.1

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

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

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 $0$ by $1$ .

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

Step 2.4.2.1.2

Multiply $- 1$ by $1$ .

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

Step 2.4.2.2

Subtract $1$ from $0$ .

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

Step 2.5

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

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Step 2.5.1

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

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

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 $2$ by $1$ .

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

Step 2.5.2.1.2

Multiply $- (- 2 \cdot 1)$ .

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Step 2.5.2.1.2.1

Multiply $- 2$ by $1$ .

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

Step 2.5.2.1.2.2

Multiply $- 1$ by $- 2$ .

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

Step 2.5.2.2

Add $2$ and $2$ .

$a⃗ \times b⃗ = - î - 1 \cdot 4 ĵ + | \begin{array}{c} 2 & 0 \\ - 2 & 1 \end{array} | k̂$

Step 2.6

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

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Step 2.6.1

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

$a⃗ \times b⃗ = - î - 1 \cdot 4 ĵ + (2 \cdot 1 - (- 2 \cdot 0)) k̂$

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 $2$ by $1$ .

$a⃗ \times b⃗ = - î - 1 \cdot 4 ĵ + (2 - (- 2 \cdot 0)) k̂$

Step 2.6.2.1.2

Multiply $- (- 2 \cdot 0)$ .

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Step 2.6.2.1.2.1

Multiply $- 2$ by $0$ .

$a⃗ \times b⃗ = - î - 1 \cdot 4 ĵ + (2 - 0) k̂$

Step 2.6.2.1.2.2

Multiply $- 1$ by $0$ .

$a⃗ \times b⃗ = - î - 1 \cdot 4 ĵ + (2 + 0) k̂$

Step 2.6.2.2

Add $2$ and $0$ .

$a⃗ \times b⃗ = - î - 1 \cdot 4 ĵ + 2 k̂$

Step 2.7

Multiply $- 1$ by $4$ .

$a⃗ \times b⃗ = - î - 4 ĵ + 2 k̂$

Step 2.8

Rewrite the answer.

$a⃗ \times b⃗ = (- 1, - 4, 2)$

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.

$| a⃗ \times b⃗ | = \sqrt{{(- 1)}^{2} + {(- 4)}^{2} + 2^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

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

$| a⃗ \times b⃗ | = \sqrt{1 + {(- 4)}^{2} + 2^{2}}$

Step 3.2.2

Raise $- 4$ to the power of $2$ .

$| a⃗ \times b⃗ | = \sqrt{1 + 16 + 2^{2}}$

Step 3.2.3

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

$| a⃗ \times b⃗ | = \sqrt{1 + 16 + 4}$

Step 3.2.4

Add $1$ and $16$ .

$| a⃗ \times b⃗ | = \sqrt{17 + 4}$

Step 3.2.5

Add $17$ and $4$ .

$| a⃗ \times b⃗ | = \sqrt{21}$

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.

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

Step 4.2

Simplify.

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Step 4.2.1

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

$| a⃗ | = \sqrt{4 + 0^{2} + 1^{2}}$

Step 4.2.2

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

$| a⃗ | = \sqrt{4 + 0 + 1^{2}}$

Step 4.2.3

One to any power is one.

$| a⃗ | = \sqrt{4 + 0 + 1}$

Step 4.2.4

Add $4$ and $0$ .

$| a⃗ | = \sqrt{4 + 1}$

Step 4.2.5

Add $4$ and $1$ .

$| a⃗ | = \sqrt{5}$

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.

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

Step 5.2

Simplify.

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Step 5.2.1

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

$| b⃗ | = \sqrt{4 + 1^{2} + 1^{2}}$

Step 5.2.2

One to any power is one.

$| b⃗ | = \sqrt{4 + 1 + 1^{2}}$

Step 5.2.3

One to any power is one.

$| b⃗ | = \sqrt{4 + 1 + 1}$

Step 5.2.4

Add $4$ and $1$ .

$| b⃗ | = \sqrt{5 + 1}$

Step 5.2.5

Add $5$ and $1$ .

$| b⃗ | = \sqrt{6}$

Step 6

Substitute the values into the formula.

$θ = \arcsin (\frac{\sqrt{21}}{\sqrt{5} \sqrt{6}})$

Step 7

Simplify.

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Step 7.1

Combine $\sqrt{21}$ and $\sqrt{6}$ into a single radical.

$θ = \arcsin (\frac{\sqrt{\frac{21}{6}}}{\sqrt{5}})$

Step 7.2

Cancel the common factor of $21$ and $6$ .

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Step 7.2.1

Factor $3$ out of $21$ .

$θ = \arcsin (\frac{\sqrt{\frac{3 (7)}{6}}}{\sqrt{5}})$

Step 7.2.2

Cancel the common factors.

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Step 7.2.2.1

Factor $3$ out of $6$ .

$θ = \arcsin (\frac{\sqrt{\frac{3 \cdot 7}{3 \cdot 2}}}{\sqrt{5}})$

Step 7.2.2.2

Cancel the common factor.

$θ = \arcsin (\frac{\sqrt{\frac{3 \cdot 7}{3 \cdot 2}}}{\sqrt{5}})$

Step 7.2.2.3

Rewrite the expression.

$θ = \arcsin (\frac{\sqrt{\frac{7}{2}}}{\sqrt{5}})$

Step 7.3

Simplify the numerator.

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Step 7.3.1

Rewrite $\sqrt{\frac{7}{2}}$ as $\frac{\sqrt{7}}{\sqrt{2}}$ .

$θ = \arcsin (\frac{\frac{\sqrt{7}}{\sqrt{2}}}{\sqrt{5}})$

Step 7.3.2

Multiply $\frac{\sqrt{7}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$θ = \arcsin (\frac{\frac{\sqrt{7}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}}}{\sqrt{5}})$

Step 7.3.3

Combine and simplify the denominator.

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Step 7.3.3.1

Multiply $\frac{\sqrt{7}}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$θ = \arcsin (\frac{\frac{\sqrt{7} \sqrt{2}}{\sqrt{2} \sqrt{2}}}{\sqrt{5}})$

Step 7.3.3.2

Raise $\sqrt{2}$ to the power of $1$ .

$θ = \arcsin (\frac{\frac{\sqrt{7} \sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}}}{\sqrt{5}})$

Step 7.3.3.3

Raise $\sqrt{2}$ to the power of $1$ .

$θ = \arcsin (\frac{\frac{\sqrt{7} \sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}}}{\sqrt{5}})$

Step 7.3.3.4

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

$θ = \arcsin (\frac{\frac{\sqrt{7} \sqrt{2}}{{\sqrt{2}}^{1 + 1}}}{\sqrt{5}})$

Step 7.3.3.5

Add $1$ and $1$ .

$θ = \arcsin (\frac{\frac{\sqrt{7} \sqrt{2}}{{\sqrt{2}}^{2}}}{\sqrt{5}})$

Step 7.3.3.6

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

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Step 7.3.3.6.1

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

$θ = \arcsin (\frac{\frac{\sqrt{7} \sqrt{2}}{{(2^{\frac{1}{2}})}^{2}}}{\sqrt{5}})$

Step 7.3.3.6.2

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

$θ = \arcsin (\frac{\frac{\sqrt{7} \sqrt{2}}{2^{\frac{1}{2} \cdot 2}}}{\sqrt{5}})$

Step 7.3.3.6.3

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

$θ = \arcsin (\frac{\frac{\sqrt{7} \sqrt{2}}{2^{\frac{2}{2}}}}{\sqrt{5}})$

Step 7.3.3.6.4

Cancel the common factor of $2$ .

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Step 7.3.3.6.4.1

Cancel the common factor.

$θ = \arcsin (\frac{\frac{\sqrt{7} \sqrt{2}}{2^{\frac{2}{2}}}}{\sqrt{5}})$

Step 7.3.3.6.4.2

Rewrite the expression.

$θ = \arcsin (\frac{\frac{\sqrt{7} \sqrt{2}}{2^{1}}}{\sqrt{5}})$

Step 7.3.3.6.5

Evaluate the exponent.

$θ = \arcsin (\frac{\frac{\sqrt{7} \sqrt{2}}{2}}{\sqrt{5}})$

Step 7.3.4

Simplify the numerator.

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Step 7.3.4.1

Combine using the product rule for radicals.

$θ = \arcsin (\frac{\frac{\sqrt{7 \cdot 2}}{2}}{\sqrt{5}})$

Step 7.3.4.2

Multiply $7$ by $2$ .

$θ = \arcsin (\frac{\frac{\sqrt{14}}{2}}{\sqrt{5}})$

Step 7.4

Multiply the numerator by the reciprocal of the denominator.

$θ = \arcsin (\frac{\sqrt{14}}{2} \cdot \frac{1}{\sqrt{5}})$

Step 7.5

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$θ = \arcsin (\frac{\sqrt{14}}{2} (\frac{1}{\sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}}))$

Step 7.6

Combine and simplify the denominator.

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Step 7.6.1

Multiply $\frac{1}{\sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$θ = \arcsin (\frac{\sqrt{14}}{2} \cdot \frac{\sqrt{5}}{\sqrt{5} \sqrt{5}})$

Step 7.6.2

Raise $\sqrt{5}$ to the power of $1$ .

$θ = \arcsin (\frac{\sqrt{14}}{2} \cdot \frac{\sqrt{5}}{{\sqrt{5}}^{1} \sqrt{5}})$

Step 7.6.3

Raise $\sqrt{5}$ to the power of $1$ .

$θ = \arcsin (\frac{\sqrt{14}}{2} \cdot \frac{\sqrt{5}}{{\sqrt{5}}^{1} {\sqrt{5}}^{1}})$

Step 7.6.4

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

$θ = \arcsin (\frac{\sqrt{14}}{2} \cdot \frac{\sqrt{5}}{{\sqrt{5}}^{1 + 1}})$

Step 7.6.5

Add $1$ and $1$ .

$θ = \arcsin (\frac{\sqrt{14}}{2} \cdot \frac{\sqrt{5}}{{\sqrt{5}}^{2}})$

Step 7.6.6

Rewrite ${\sqrt{5}}^{2}$ as $5$ .

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Step 7.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5}$ as $5^{\frac{1}{2}}$ .

$θ = \arcsin (\frac{\sqrt{14}}{2} \cdot \frac{\sqrt{5}}{{(5^{\frac{1}{2}})}^{2}})$

Step 7.6.6.2

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

$θ = \arcsin (\frac{\sqrt{14}}{2} \cdot \frac{\sqrt{5}}{5^{\frac{1}{2} \cdot 2}})$

Step 7.6.6.3

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

$θ = \arcsin (\frac{\sqrt{14}}{2} \cdot \frac{\sqrt{5}}{5^{\frac{2}{2}}})$

Step 7.6.6.4

Cancel the common factor of $2$ .

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Step 7.6.6.4.1

Cancel the common factor.

$θ = \arcsin (\frac{\sqrt{14}}{2} \cdot \frac{\sqrt{5}}{5^{\frac{2}{2}}})$

Step 7.6.6.4.2

Rewrite the expression.

$θ = \arcsin (\frac{\sqrt{14}}{2} \cdot \frac{\sqrt{5}}{5^{1}})$

Step 7.6.6.5

Evaluate the exponent.

$θ = \arcsin (\frac{\sqrt{14}}{2} \cdot \frac{\sqrt{5}}{5})$

Step 7.7

Multiply $\frac{\sqrt{14}}{2} \cdot \frac{\sqrt{5}}{5}$ .

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Step 7.7.1

Multiply $\frac{\sqrt{14}}{2}$ by $\frac{\sqrt{5}}{5}$ .

$θ = \arcsin (\frac{\sqrt{14} \sqrt{5}}{2 \cdot 5})$

Step 7.7.2

Combine using the product rule for radicals.

$θ = \arcsin (\frac{\sqrt{14 \cdot 5}}{2 \cdot 5})$

Step 7.7.3

Multiply $14$ by $5$ .

$θ = \arcsin (\frac{\sqrt{70}}{2 \cdot 5})$

Step 7.7.4

Multiply $2$ by $5$ .

$θ = \arcsin (\frac{\sqrt{70}}{10})$

Step 7.8

Evaluate $\arcsin (\frac{\sqrt{70}}{10})$ .

$θ = 56.78908923$

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