Precalculus Examples

$(2, 4)$ , $(- 4, 9)$

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.

$a⃗ \cdot b⃗ = 2 \cdot - 4 + 4 \cdot 9$

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 $- 4$ .

$a⃗ \cdot b⃗ = - 8 + 4 \cdot 9$

Step 2.2.1.2

Multiply $4$ by $9$ .

$a⃗ \cdot b⃗ = - 8 + 36$

Step 2.2.2

Add $- 8$ and $36$ .

$a⃗ \cdot b⃗ = 28$

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.

$| a⃗ | = \sqrt{2^{2} + 4^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

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

$| a⃗ | = \sqrt{4 + 4^{2}}$

Step 3.2.2

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

$| a⃗ | = \sqrt{4 + 16}$

Step 3.2.3

Add $4$ and $16$ .

$| a⃗ | = \sqrt{20}$

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.

$| b⃗ | = \sqrt{{(- 4)}^{2} + 9^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$| b⃗ | = \sqrt{16 + 9^{2}}$

Step 4.2.2

Raise $9$ to the power of $2$ .

$| b⃗ | = \sqrt{16 + 81}$

Step 4.2.3

Add $16$ and $81$ .

$| b⃗ | = \sqrt{97}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{28}{2 \sqrt{5} \sqrt{97}})$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $28$ and $2$ .

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Step 6.1.1

Factor $2$ out of $28$ .

$θ = \arccos (\frac{2 \cdot 14}{2 \sqrt{5} \sqrt{97}})$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $2$ out of $2 \sqrt{5} \sqrt{97}$ .

$θ = \arccos (\frac{2 \cdot 14}{2 (\sqrt{5} \sqrt{97})})$

Step 6.1.2.2

Cancel the common factor.

$θ = \arccos (\frac{2 \cdot 14}{2 (\sqrt{5} \sqrt{97})})$

Step 6.1.2.3

Rewrite the expression.

$θ = \arccos (\frac{14}{\sqrt{5} \sqrt{97}})$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Combine using the product rule for radicals.

$θ = \arccos (\frac{14}{\sqrt{5 \cdot 97}})$

Step 6.2.2

Multiply $5$ by $97$ .

$θ = \arccos (\frac{14}{\sqrt{485}})$

Step 6.3

Multiply $\frac{14}{\sqrt{485}}$ by $\frac{\sqrt{485}}{\sqrt{485}}$ .

$θ = \arccos (\frac{14}{\sqrt{485}} \cdot \frac{\sqrt{485}}{\sqrt{485}})$

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

Multiply $\frac{14}{\sqrt{485}}$ by $\frac{\sqrt{485}}{\sqrt{485}}$ .

$θ = \arccos (\frac{14 \sqrt{485}}{\sqrt{485} \sqrt{485}})$

Step 6.4.2

Raise $\sqrt{485}$ to the power of $1$ .

$θ = \arccos (\frac{14 \sqrt{485}}{{\sqrt{485}}^{1} \sqrt{485}})$

Step 6.4.3

Raise $\sqrt{485}$ to the power of $1$ .

$θ = \arccos (\frac{14 \sqrt{485}}{{\sqrt{485}}^{1} {\sqrt{485}}^{1}})$

Step 6.4.4

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

$θ = \arccos (\frac{14 \sqrt{485}}{{\sqrt{485}}^{1 + 1}})$

Step 6.4.5

Add $1$ and $1$ .

$θ = \arccos (\frac{14 \sqrt{485}}{{\sqrt{485}}^{2}})$

Step 6.4.6

Rewrite ${\sqrt{485}}^{2}$ as $485$ .

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Step 6.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{485}$ as $485^{\frac{1}{2}}$ .

$θ = \arccos (\frac{14 \sqrt{485}}{{(485^{\frac{1}{2}})}^{2}})$

Step 6.4.6.2

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

$θ = \arccos (\frac{14 \sqrt{485}}{485^{\frac{1}{2} \cdot 2}})$

Step 6.4.6.3

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

$θ = \arccos (\frac{14 \sqrt{485}}{485^{\frac{2}{2}}})$

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.

$θ = \arccos (\frac{14 \sqrt{485}}{485^{\frac{2}{2}}})$

Step 6.4.6.4.2

Rewrite the expression.

$θ = \arccos (\frac{14 \sqrt{485}}{485^{1}})$

Step 6.4.6.5

Evaluate the exponent.

$θ = \arccos (\frac{14 \sqrt{485}}{485})$

Step 6.5

Evaluate $\arccos (\frac{14 \sqrt{485}}{485})$ .

$θ = 50.52754015$

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