Pre-Algebra Examples

$(4, - 2)$ , $(5, - 5)$

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.

$a⃗ \cdot b⃗ = 4 \cdot 5 - 2 \cdot - 5$

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 $4$ by $5$ .

$a⃗ \cdot b⃗ = 20 - 2 \cdot - 5$

Step 2.2.1.2

Multiply $- 2$ by $- 5$ .

$a⃗ \cdot b⃗ = 20 + 10$

Step 2.2.2

Add $20$ and $10$ .

$a⃗ \cdot b⃗ = 30$

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.

$| a⃗ | = \sqrt{4^{2} + {(- 2)}^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

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

$| a⃗ | = \sqrt{16 + {(- 2)}^{2}}$

Step 3.2.2

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

$| a⃗ | = \sqrt{16 + 4}$

Step 3.2.3

Add $16$ and $4$ .

$| a⃗ | = \sqrt{20}$

Step 3.2.4

Rewrite $20$ as $2^{2} \cdot 5$ .

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Step 3.2.4.1

Factor $4$ out of $20$ .

$| a⃗ | = \sqrt{4 (5)}$

Step 3.2.4.2

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

$| a⃗ | = \sqrt{2^{2} \cdot 5}$

Step 3.2.5

Pull terms out from under the radical.

$| a⃗ | = 2 \sqrt{5}$

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.

$| b⃗ | = \sqrt{5^{2} + {(- 5)}^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$| b⃗ | = \sqrt{25 + {(- 5)}^{2}}$

Step 4.2.2

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

$| b⃗ | = \sqrt{25 + 25}$

Step 4.2.3

Add $25$ and $25$ .

$| b⃗ | = \sqrt{50}$

Step 4.2.4

Rewrite $50$ as $5^{2} \cdot 2$ .

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Step 4.2.4.1

Factor $25$ out of $50$ .

$| b⃗ | = \sqrt{25 (2)}$

Step 4.2.4.2

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

$| b⃗ | = \sqrt{5^{2} \cdot 2}$

Step 4.2.5

Pull terms out from under the radical.

$| b⃗ | = 5 \sqrt{2}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{30}{2 \sqrt{5} (5 \sqrt{2})})$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $30$ and $2$ .

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Step 6.1.1

Factor $2$ out of $30$ .

$θ = \arccos (\frac{2 \cdot 15}{2 \sqrt{5} (5 \sqrt{2})})$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

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

$θ = \arccos (\frac{2 \cdot 15}{2 (\sqrt{5} (5 \sqrt{2}))})$

Step 6.1.2.2

Cancel the common factor.

$θ = \arccos (\frac{2 \cdot 15}{2 (\sqrt{5} (5 \sqrt{2}))})$

Step 6.1.2.3

Rewrite the expression.

$θ = \arccos (\frac{15}{\sqrt{5} (5 \sqrt{2})})$

Step 6.2

Cancel the common factor of $15$ and $5$ .

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Step 6.2.1

Factor $5$ out of $15$ .

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

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

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

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

Step 6.2.2.2

Cancel the common factor.

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

Step 6.2.2.3

Rewrite the expression.

$θ = \arccos (\frac{3}{\sqrt{5} (\sqrt{2})})$

Step 6.3

Simplify the denominator.

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Step 6.3.1

Combine using the product rule for radicals.

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

Step 6.3.2

Multiply $5$ by $2$ .

$θ = \arccos (\frac{3}{\sqrt{10}})$

Step 6.4

Multiply $\frac{3}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$θ = \arccos (\frac{3}{\sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}})$

Step 6.5

Combine and simplify the denominator.

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Step 6.5.1

Multiply $\frac{3}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$θ = \arccos (\frac{3 \sqrt{10}}{\sqrt{10} \sqrt{10}})$

Step 6.5.2

Raise $\sqrt{10}$ to the power of $1$ .

$θ = \arccos (\frac{3 \sqrt{10}}{{\sqrt{10}}^{1} \sqrt{10}})$

Step 6.5.3

Raise $\sqrt{10}$ to the power of $1$ .

$θ = \arccos (\frac{3 \sqrt{10}}{{\sqrt{10}}^{1} {\sqrt{10}}^{1}})$

Step 6.5.4

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

$θ = \arccos (\frac{3 \sqrt{10}}{{\sqrt{10}}^{1 + 1}})$

Step 6.5.5

Add $1$ and $1$ .

$θ = \arccos (\frac{3 \sqrt{10}}{{\sqrt{10}}^{2}})$

Step 6.5.6

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

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Step 6.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$θ = \arccos (\frac{3 \sqrt{10}}{{(10^{\frac{1}{2}})}^{2}})$

Step 6.5.6.2

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

$θ = \arccos (\frac{3 \sqrt{10}}{10^{\frac{1}{2} \cdot 2}})$

Step 6.5.6.3

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

$θ = \arccos (\frac{3 \sqrt{10}}{10^{\frac{2}{2}}})$

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.

$θ = \arccos (\frac{3 \sqrt{10}}{10^{\frac{2}{2}}})$

Step 6.5.6.4.2

Rewrite the expression.

$θ = \arccos (\frac{3 \sqrt{10}}{10^{1}})$

Step 6.5.6.5

Evaluate the exponent.

$θ = \arccos (\frac{3 \sqrt{10}}{10})$

Step 6.6

Evaluate $\arccos (\frac{3 \sqrt{10}}{10})$ .

$θ = 18.43494882$