Pre-Algebra Examples

$(2, - 6)$ , $(5, 0)$

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.

$a⃗ \cdot b⃗ = 2 \cdot 5 - 6 \cdot 0$

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 $5$ .

$a⃗ \cdot b⃗ = 10 - 6 \cdot 0$

Step 2.2.1.2

Multiply $- 6$ by $0$ .

$a⃗ \cdot b⃗ = 10 + 0$

Step 2.2.2

Add $10$ and $0$ .

$a⃗ \cdot b⃗ = 10$

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.

$| a⃗ | = \sqrt{2^{2} + {(- 6)}^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

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

$| a⃗ | = \sqrt{4 + {(- 6)}^{2}}$

Step 3.2.2

Raise $- 6$ to the power of $2$ .

$| a⃗ | = \sqrt{4 + 36}$

Step 3.2.3

Add $4$ and $36$ .

$| a⃗ | = \sqrt{40}$

Step 3.2.4

Rewrite $40$ as $2^{2} \cdot 10$ .

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Step 3.2.4.1

Factor $4$ out of $40$ .

$| a⃗ | = \sqrt{4 (10)}$

Step 3.2.4.2

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

$| a⃗ | = \sqrt{2^{2} \cdot 10}$

Step 3.2.5

Pull terms out from under the radical.

$| a⃗ | = 2 \sqrt{10}$

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.

$| b⃗ | = \sqrt{5^{2} + 0^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$| b⃗ | = \sqrt{25 + 0^{2}}$

Step 4.2.2

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

$| b⃗ | = \sqrt{25 + 0}$

Step 4.2.3

Add $25$ and $0$ .

$| b⃗ | = \sqrt{25}$

Step 4.2.4

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

$| b⃗ | = \sqrt{5^{2}}$

Step 4.2.5

Pull terms out from under the radical, assuming positive real numbers.

$| b⃗ | = 5$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{10}{2 \sqrt{10} \cdot 5})$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $10$ and $2$ .

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Step 6.1.1

Factor $2$ out of $10$ .

$θ = \arccos (\frac{2 \cdot 5}{2 \sqrt{10} \cdot 5})$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $2$ out of $2 \sqrt{10} \cdot 5$ .

$θ = \arccos (\frac{2 \cdot 5}{2 (\sqrt{10} \cdot 5)})$

Step 6.1.2.2

Cancel the common factor.

$θ = \arccos (\frac{2 \cdot 5}{2 (\sqrt{10} \cdot 5)})$

Step 6.1.2.3

Rewrite the expression.

$θ = \arccos (\frac{5}{\sqrt{10} \cdot 5})$

Step 6.2

Cancel the common factor of $5$ .

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Step 6.2.1

Cancel the common factor.

$θ = \arccos (\frac{5}{\sqrt{10} \cdot 5})$

Step 6.2.2

Rewrite the expression.

$θ = \arccos (\frac{1}{\sqrt{10}})$

Step 6.3

Multiply $\frac{1}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

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

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

Multiply $\frac{1}{\sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

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

Step 6.4.2

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

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

Step 6.4.3

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

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

Step 6.4.4

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

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

Step 6.4.5

Add $1$ and $1$ .

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

Step 6.4.6

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

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Step 6.4.6.1

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

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

Step 6.4.6.2

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

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

Step 6.4.6.3

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

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

Step 6.4.6.4

Cancel the common factor of $2$ .

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Step 6.4.6.4.1

Cancel the common factor.

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

Step 6.4.6.4.2

Rewrite the expression.

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

Step 6.4.6.5

Evaluate the exponent.

$θ = \arccos (\frac{\sqrt{10}}{10})$

Step 6.5

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

$θ = 71.56505117$