Precalculus Examples

$(6, 8)$ , $(2, 4)$

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.

$a⃗ \cdot b⃗ = 6 \cdot 2 + 8 \cdot 4$

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 $6$ by $2$ .

$a⃗ \cdot b⃗ = 12 + 8 \cdot 4$

Step 2.2.1.2

Multiply $8$ by $4$ .

$a⃗ \cdot b⃗ = 12 + 32$

Step 2.2.2

Add $12$ and $32$ .

$a⃗ \cdot b⃗ = 44$

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.

$| a⃗ | = \sqrt{6^{2} + 8^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

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

$| a⃗ | = \sqrt{36 + 8^{2}}$

Step 3.2.2

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

$| a⃗ | = \sqrt{36 + 64}$

Step 3.2.3

Add $36$ and $64$ .

$| a⃗ | = \sqrt{100}$

Step 3.2.4

Rewrite $100$ as $10^{2}$ .

$| a⃗ | = \sqrt{10^{2}}$

Step 3.2.5

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

$| a⃗ | = 10$

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.

$| b⃗ | = \sqrt{2^{2} + 4^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$| b⃗ | = \sqrt{4 + 4^{2}}$

Step 4.2.2

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

$| b⃗ | = \sqrt{4 + 16}$

Step 4.2.3

Add $4$ and $16$ .

$| b⃗ | = \sqrt{20}$

Step 4.2.4

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

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Step 4.2.4.1

Factor $4$ out of $20$ .

$| b⃗ | = \sqrt{4 (5)}$

Step 4.2.4.2

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

$| b⃗ | = \sqrt{2^{2} \cdot 5}$

Step 4.2.5

Pull terms out from under the radical.

$| b⃗ | = 2 \sqrt{5}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{44}{10 (2 \sqrt{5})})$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $44$ and $10$ .

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Step 6.1.1

Factor $2$ out of $44$ .

$θ = \arccos (\frac{2 (22)}{10 (2 \sqrt{5})})$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

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

$θ = \arccos (\frac{2 (22)}{2 (5 (2 \sqrt{5}))})$

Step 6.1.2.2

Cancel the common factor.

$θ = \arccos (\frac{2 \cdot 22}{2 (5 (2 \sqrt{5}))})$

Step 6.1.2.3

Rewrite the expression.

$θ = \arccos (\frac{22}{5 (2 \sqrt{5})})$

Step 6.2

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

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Step 6.2.1

Factor $2$ out of $22$ .

$θ = \arccos (\frac{2 \cdot 11}{5 (2 \sqrt{5})})$

Step 6.2.2

Cancel the common factors.

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Step 6.2.2.1

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

$θ = \arccos (\frac{2 \cdot 11}{2 (5 (\sqrt{5}))})$

Step 6.2.2.2

Cancel the common factor.

$θ = \arccos (\frac{2 \cdot 11}{2 (5 (\sqrt{5}))})$

Step 6.2.2.3

Rewrite the expression.

$θ = \arccos (\frac{11}{5 (\sqrt{5})})$

Step 6.3

Multiply $\frac{11}{5 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$θ = \arccos (\frac{11}{5 \sqrt{5}} \cdot \frac{\sqrt{5}}{\sqrt{5}})$

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

Multiply $\frac{11}{5 \sqrt{5}}$ by $\frac{\sqrt{5}}{\sqrt{5}}$ .

$θ = \arccos (\frac{11 \sqrt{5}}{5 \sqrt{5} \sqrt{5}})$

Step 6.4.2

Move $\sqrt{5}$ .

$θ = \arccos (\frac{11 \sqrt{5}}{5 (\sqrt{5} \sqrt{5})})$

Step 6.4.3

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

$θ = \arccos (\frac{11 \sqrt{5}}{5 ({\sqrt{5}}^{1} \sqrt{5})})$

Step 6.4.4

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

$θ = \arccos (\frac{11 \sqrt{5}}{5 ({\sqrt{5}}^{1} {\sqrt{5}}^{1})})$

Step 6.4.5

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

$θ = \arccos (\frac{11 \sqrt{5}}{5 {\sqrt{5}}^{1 + 1}})$

Step 6.4.6

Add $1$ and $1$ .

$θ = \arccos (\frac{11 \sqrt{5}}{5 {\sqrt{5}}^{2}})$

Step 6.4.7

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

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Step 6.4.7.1

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

$θ = \arccos (\frac{11 \sqrt{5}}{5 {(5^{\frac{1}{2}})}^{2}})$

Step 6.4.7.2

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

$θ = \arccos (\frac{11 \sqrt{5}}{5 \cdot 5^{\frac{1}{2} \cdot 2}})$

Step 6.4.7.3

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

$θ = \arccos (\frac{11 \sqrt{5}}{5 \cdot 5^{\frac{2}{2}}})$

Step 6.4.7.4

Cancel the common factor of $2$ .

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Step 6.4.7.4.1

Cancel the common factor.

$θ = \arccos (\frac{11 \sqrt{5}}{5 \cdot 5^{\frac{2}{2}}})$

Step 6.4.7.4.2

Rewrite the expression.

$θ = \arccos (\frac{11 \sqrt{5}}{5 \cdot 5^{1}})$

Step 6.4.7.5

Evaluate the exponent.

$θ = \arccos (\frac{11 \sqrt{5}}{5 \cdot 5})$

Step 6.5

Multiply $5$ by $5$ .

$θ = \arccos (\frac{11 \sqrt{5}}{25})$

Step 6.6

Evaluate $\arccos (\frac{11 \sqrt{5}}{25})$ .

$θ = 10.30484646$

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