Pre-Algebra Examples

$(- 3, 8)$ , $(2, - 2)$

Step 1

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

$θ = \arccos (\frac{a⃗ \cdot b⃗}{| a⃗ | | b⃗ |})$

Step 2

Find the dot product.

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Step 2.1

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

$a⃗ \cdot b⃗ = - 3 \cdot 2 + 8 \cdot - 2$

Step 2.2

Simplify.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

Multiply $- 3$ by $2$ .

$a⃗ \cdot b⃗ = - 6 + 8 \cdot - 2$

Step 2.2.1.2

Multiply $8$ by $- 2$ .

$a⃗ \cdot b⃗ = - 6 - 16$

Step 2.2.2

Subtract $16$ from $- 6$ .

$a⃗ \cdot b⃗ = - 22$

Step 3

Find the magnitude of $a⃗$ .

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Step 3.1

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

$| a⃗ | = \sqrt{{(- 3)}^{2} + 8^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

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

$| a⃗ | = \sqrt{9 + 8^{2}}$

Step 3.2.2

Raise $8$ to the power of $2$ .

$| a⃗ | = \sqrt{9 + 64}$

Step 3.2.3

Add $9$ and $64$ .

$| a⃗ | = \sqrt{73}$

Step 4

Find the magnitude of $b⃗$ .

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Step 4.1

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

$| b⃗ | = \sqrt{2^{2} + {(- 2)}^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$| b⃗ | = \sqrt{4 + {(- 2)}^{2}}$

Step 4.2.2

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

$| b⃗ | = \sqrt{4 + 4}$

Step 4.2.3

Add $4$ and $4$ .

$| b⃗ | = \sqrt{8}$

Step 4.2.4

Rewrite $8$ as $2^{2} \cdot 2$ .

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Step 4.2.4.1

Factor $4$ out of $8$ .

$| b⃗ | = \sqrt{4 (2)}$

Step 4.2.4.2

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

$| b⃗ | = \sqrt{2^{2} \cdot 2}$

Step 4.2.5

Pull terms out from under the radical.

$| b⃗ | = 2 \sqrt{2}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{- 22}{\sqrt{73} (2 \sqrt{2})})$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $- 22$ and $2$ .

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Step 6.1.1

Factor $2$ out of $- 22$ .

$θ = \arccos (\frac{2 \cdot - 11}{\sqrt{73} (2 \sqrt{2})})$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $2$ out of $\sqrt{73} (2 \sqrt{2})$ .

$θ = \arccos (\frac{2 \cdot - 11}{2 (\sqrt{73} (\sqrt{2}))})$

Step 6.1.2.2

Cancel the common factor.

$θ = \arccos (\frac{2 \cdot - 11}{2 (\sqrt{73} (\sqrt{2}))})$

Step 6.1.2.3

Rewrite the expression.

$θ = \arccos (\frac{- 11}{\sqrt{73} (\sqrt{2})})$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Combine using the product rule for radicals.

$θ = \arccos (\frac{- 11}{\sqrt{73 \cdot 2}})$

Step 6.2.2

Multiply $73$ by $2$ .

$θ = \arccos (\frac{- 11}{\sqrt{146}})$

Step 6.3

Move the negative in front of the fraction.

$θ = \arccos (- \frac{11}{\sqrt{146}})$

Step 6.4

Multiply $\frac{11}{\sqrt{146}}$ by $\frac{\sqrt{146}}{\sqrt{146}}$ .

$θ = \arccos (- (\frac{11}{\sqrt{146}} \cdot \frac{\sqrt{146}}{\sqrt{146}}))$

Step 6.5

Combine and simplify the denominator.

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Step 6.5.1

Multiply $\frac{11}{\sqrt{146}}$ by $\frac{\sqrt{146}}{\sqrt{146}}$ .

$θ = \arccos (- \frac{11 \sqrt{146}}{\sqrt{146} \sqrt{146}})$

Step 6.5.2

Raise $\sqrt{146}$ to the power of $1$ .

$θ = \arccos (- \frac{11 \sqrt{146}}{{\sqrt{146}}^{1} \sqrt{146}})$

Step 6.5.3

Raise $\sqrt{146}$ to the power of $1$ .

$θ = \arccos (- \frac{11 \sqrt{146}}{{\sqrt{146}}^{1} {\sqrt{146}}^{1}})$

Step 6.5.4

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

$θ = \arccos (- \frac{11 \sqrt{146}}{{\sqrt{146}}^{1 + 1}})$

Step 6.5.5

Add $1$ and $1$ .

$θ = \arccos (- \frac{11 \sqrt{146}}{{\sqrt{146}}^{2}})$

Step 6.5.6

Rewrite ${\sqrt{146}}^{2}$ as $146$ .

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Step 6.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{146}$ as $146^{\frac{1}{2}}$ .

$θ = \arccos (- \frac{11 \sqrt{146}}{{(146^{\frac{1}{2}})}^{2}})$

Step 6.5.6.2

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

$θ = \arccos (- \frac{11 \sqrt{146}}{146^{\frac{1}{2} \cdot 2}})$

Step 6.5.6.3

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

$θ = \arccos (- \frac{11 \sqrt{146}}{146^{\frac{2}{2}}})$

Step 6.5.6.4

Cancel the common factor of $2$ .

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Step 6.5.6.4.1

Cancel the common factor.

$θ = \arccos (- \frac{11 \sqrt{146}}{146^{\frac{2}{2}}})$

Step 6.5.6.4.2

Rewrite the expression.

$θ = \arccos (- \frac{11 \sqrt{146}}{146^{1}})$

Step 6.5.6.5

Evaluate the exponent.

$θ = \arccos (- \frac{11 \sqrt{146}}{146})$

Step 6.6

Evaluate $\arccos (- \frac{11 \sqrt{146}}{146})$ .

$θ = 155.55604521$