Pre-Algebra Examples

$(- 2, - 24)$ , $(5, 25)$

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.

$a⃗ \cdot b⃗ = - 2 \cdot 5 - 24 \cdot 25$

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 $- 2$ by $5$ .

$a⃗ \cdot b⃗ = - 10 - 24 \cdot 25$

Step 2.2.1.2

Multiply $- 24$ by $25$ .

$a⃗ \cdot b⃗ = - 10 - 600$

Step 2.2.2

Subtract $600$ from $- 10$ .

$a⃗ \cdot b⃗ = - 610$

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.

$| a⃗ | = \sqrt{{(- 2)}^{2} + {(- 24)}^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

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

$| a⃗ | = \sqrt{4 + {(- 24)}^{2}}$

Step 3.2.2

Raise $- 24$ to the power of $2$ .

$| a⃗ | = \sqrt{4 + 576}$

Step 3.2.3

Add $4$ and $576$ .

$| a⃗ | = \sqrt{580}$

Step 3.2.4

Rewrite $580$ as $2^{2} \cdot 145$ .

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Step 3.2.4.1

Factor $4$ out of $580$ .

$| a⃗ | = \sqrt{4 (145)}$

Step 3.2.4.2

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

$| a⃗ | = \sqrt{2^{2} \cdot 145}$

Step 3.2.5

Pull terms out from under the radical.

$| a⃗ | = 2 \sqrt{145}$

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.

$| b⃗ | = \sqrt{5^{2} + 25^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

Raise $5$ to the power of $2$ .

$| b⃗ | = \sqrt{25 + 25^{2}}$

Step 4.2.2

Raise $25$ to the power of $2$ .

$| b⃗ | = \sqrt{25 + 625}$

Step 4.2.3

Add $25$ and $625$ .

$| b⃗ | = \sqrt{650}$

Step 4.2.4

Rewrite $650$ as $5^{2} \cdot 26$ .

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Step 4.2.4.1

Factor $25$ out of $650$ .

$| b⃗ | = \sqrt{25 (26)}$

Step 4.2.4.2

Rewrite $25$ as $5^{2}$ .

$| b⃗ | = \sqrt{5^{2} \cdot 26}$

Step 4.2.5

Pull terms out from under the radical.

$| b⃗ | = 5 \sqrt{26}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{- 610}{2 \sqrt{145} (5 \sqrt{26})})$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $- 610$ and $2$ .

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Step 6.1.1

Factor $2$ out of $- 610$ .

$θ = \arccos (\frac{2 \cdot - 305}{2 \sqrt{145} (5 \sqrt{26})})$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $2$ out of $2 \sqrt{145} (5 \sqrt{26})$ .

$θ = \arccos (\frac{2 \cdot - 305}{2 (\sqrt{145} (5 \sqrt{26}))})$

Step 6.1.2.2

Cancel the common factor.

$θ = \arccos (\frac{2 \cdot - 305}{2 (\sqrt{145} (5 \sqrt{26}))})$

Step 6.1.2.3

Rewrite the expression.

$θ = \arccos (\frac{- 305}{\sqrt{145} (5 \sqrt{26})})$

Step 6.2

Cancel the common factor of $- 305$ and $5$ .

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Step 6.2.1

Factor $5$ out of $- 305$ .

$θ = \arccos (\frac{5 \cdot - 61}{\sqrt{145} (5 \sqrt{26})})$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

Factor $5$ out of $\sqrt{145} (5 \sqrt{26})$ .

$θ = \arccos (\frac{5 \cdot - 61}{5 (\sqrt{145} (\sqrt{26}))})$

Step 6.2.2.2

Cancel the common factor.

$θ = \arccos (\frac{5 \cdot - 61}{5 (\sqrt{145} (\sqrt{26}))})$

Step 6.2.2.3

Rewrite the expression.

$θ = \arccos (\frac{- 61}{\sqrt{145} (\sqrt{26})})$

Step 6.3

Simplify the denominator.

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Step 6.3.1

Combine using the product rule for radicals.

$θ = \arccos (\frac{- 61}{\sqrt{145 \cdot 26}})$

Step 6.3.2

Multiply $145$ by $26$ .

$θ = \arccos (\frac{- 61}{\sqrt{3770}})$

Step 6.4

Move the negative in front of the fraction.

$θ = \arccos (- \frac{61}{\sqrt{3770}})$

Step 6.5

Multiply $\frac{61}{\sqrt{3770}}$ by $\frac{\sqrt{3770}}{\sqrt{3770}}$ .

$θ = \arccos (- (\frac{61}{\sqrt{3770}} \cdot \frac{\sqrt{3770}}{\sqrt{3770}}))$

Step 6.6

Combine and simplify the denominator.

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Step 6.6.1

Multiply $\frac{61}{\sqrt{3770}}$ by $\frac{\sqrt{3770}}{\sqrt{3770}}$ .

$θ = \arccos (- \frac{61 \sqrt{3770}}{\sqrt{3770} \sqrt{3770}})$

Step 6.6.2

Raise $\sqrt{3770}$ to the power of $1$ .

$θ = \arccos (- \frac{61 \sqrt{3770}}{{\sqrt{3770}}^{1} \sqrt{3770}})$

Step 6.6.3

Raise $\sqrt{3770}$ to the power of $1$ .

$θ = \arccos (- \frac{61 \sqrt{3770}}{{\sqrt{3770}}^{1} {\sqrt{3770}}^{1}})$

Step 6.6.4

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

$θ = \arccos (- \frac{61 \sqrt{3770}}{{\sqrt{3770}}^{1 + 1}})$

Step 6.6.5

Add $1$ and $1$ .

$θ = \arccos (- \frac{61 \sqrt{3770}}{{\sqrt{3770}}^{2}})$

Step 6.6.6

Rewrite ${\sqrt{3770}}^{2}$ as $3770$ .

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Step 6.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3770}$ as $3770^{\frac{1}{2}}$ .

$θ = \arccos (- \frac{61 \sqrt{3770}}{{(3770^{\frac{1}{2}})}^{2}})$

Step 6.6.6.2

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

$θ = \arccos (- \frac{61 \sqrt{3770}}{3770^{\frac{1}{2} \cdot 2}})$

Step 6.6.6.3

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

$θ = \arccos (- \frac{61 \sqrt{3770}}{3770^{\frac{2}{2}}})$

Step 6.6.6.4

Cancel the common factor of $2$ .

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Step 6.6.6.4.1

Cancel the common factor.

$θ = \arccos (- \frac{61 \sqrt{3770}}{3770^{\frac{2}{2}}})$

Step 6.6.6.4.2

Rewrite the expression.

$θ = \arccos (- \frac{61 \sqrt{3770}}{3770^{1}})$

Step 6.6.6.5

Evaluate the exponent.

$θ = \arccos (- \frac{61 \sqrt{3770}}{3770})$

Step 6.7

Evaluate $\arccos (- \frac{61 \sqrt{3770}}{3770})$ .

$θ = 173.45370921$