Precalculus Examples

$(1, 2)$ , $(2, 9)$

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.

$a⃗ \cdot b⃗ = 1 \cdot 2 + 2 \cdot 9$

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 $1$ .

$a⃗ \cdot b⃗ = 2 + 2 \cdot 9$

Step 2.2.1.2

Multiply $2$ by $9$ .

$a⃗ \cdot b⃗ = 2 + 18$

Step 2.2.2

Add $2$ and $18$ .

$a⃗ \cdot b⃗ = 20$

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.

$| a⃗ | = \sqrt{1^{2} + 2^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

One to any power is one.

$| a⃗ | = \sqrt{1 + 2^{2}}$

Step 3.2.2

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

$| a⃗ | = \sqrt{1 + 4}$

Step 3.2.3

Add $1$ and $4$ .

$| a⃗ | = \sqrt{5}$

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.

$| b⃗ | = \sqrt{2^{2} + 9^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$| b⃗ | = \sqrt{4 + 9^{2}}$

Step 4.2.2

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

$| b⃗ | = \sqrt{4 + 81}$

Step 4.2.3

Add $4$ and $81$ .

$| b⃗ | = \sqrt{85}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{20}{\sqrt{5} \sqrt{85}})$

Step 6

Simplify.

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Step 6.1

Simplify the denominator.

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Step 6.1.1

Combine using the product rule for radicals.

$θ = \arccos (\frac{20}{\sqrt{5 \cdot 85}})$

Step 6.1.2

Multiply $5$ by $85$ .

$θ = \arccos (\frac{20}{\sqrt{425}})$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Rewrite $425$ as $5^{2} \cdot 17$ .

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Step 6.2.1.1

Factor $25$ out of $425$ .

$θ = \arccos (\frac{20}{\sqrt{25 (17)}})$

Step 6.2.1.2

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

$θ = \arccos (\frac{20}{\sqrt{5^{2} \cdot 17}})$

Step 6.2.2

Pull terms out from under the radical.

$θ = \arccos (\frac{20}{5 \sqrt{17}})$

Step 6.3

Cancel the common factor of $20$ and $5$ .

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Step 6.3.1

Factor $5$ out of $20$ .

$θ = \arccos (\frac{5 \cdot 4}{5 \sqrt{17}})$

Step 6.3.2

Cancel the common factors.

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Step 6.3.2.1

Factor $5$ out of $5 \sqrt{17}$ .

$θ = \arccos (\frac{5 \cdot 4}{5 (\sqrt{17})})$

Step 6.3.2.2

Cancel the common factor.

$θ = \arccos (\frac{5 \cdot 4}{5 \sqrt{17}})$

Step 6.3.2.3

Rewrite the expression.

$θ = \arccos (\frac{4}{\sqrt{17}})$

Step 6.4

Multiply $\frac{4}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$θ = \arccos (\frac{4}{\sqrt{17}} \cdot \frac{\sqrt{17}}{\sqrt{17}})$

Step 6.5

Combine and simplify the denominator.

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Step 6.5.1

Multiply $\frac{4}{\sqrt{17}}$ by $\frac{\sqrt{17}}{\sqrt{17}}$ .

$θ = \arccos (\frac{4 \sqrt{17}}{\sqrt{17} \sqrt{17}})$

Step 6.5.2

Raise $\sqrt{17}$ to the power of $1$ .

$θ = \arccos (\frac{4 \sqrt{17}}{{\sqrt{17}}^{1} \sqrt{17}})$

Step 6.5.3

Raise $\sqrt{17}$ to the power of $1$ .

$θ = \arccos (\frac{4 \sqrt{17}}{{\sqrt{17}}^{1} {\sqrt{17}}^{1}})$

Step 6.5.4

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

$θ = \arccos (\frac{4 \sqrt{17}}{{\sqrt{17}}^{1 + 1}})$

Step 6.5.5

Add $1$ and $1$ .

$θ = \arccos (\frac{4 \sqrt{17}}{{\sqrt{17}}^{2}})$

Step 6.5.6

Rewrite ${\sqrt{17}}^{2}$ as $17$ .

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Step 6.5.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{17}$ as $17^{\frac{1}{2}}$ .

$θ = \arccos (\frac{4 \sqrt{17}}{{(17^{\frac{1}{2}})}^{2}})$

Step 6.5.6.2

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

$θ = \arccos (\frac{4 \sqrt{17}}{17^{\frac{1}{2} \cdot 2}})$

Step 6.5.6.3

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

$θ = \arccos (\frac{4 \sqrt{17}}{17^{\frac{2}{2}}})$

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.

$θ = \arccos (\frac{4 \sqrt{17}}{17^{\frac{2}{2}}})$

Step 6.5.6.4.2

Rewrite the expression.

$θ = \arccos (\frac{4 \sqrt{17}}{17^{1}})$

Step 6.5.6.5

Evaluate the exponent.

$θ = \arccos (\frac{4 \sqrt{17}}{17})$

Step 6.6

Evaluate $\arccos (\frac{4 \sqrt{17}}{17})$ .

$θ = 14.03624346$

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