Examples

$(3, 8)$ , $(9, 12)$

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.

$a⃗ \cdot b⃗ = 3 \cdot 9 + 8 \cdot 12$

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 $3$ by $9$ .

$a⃗ \cdot b⃗ = 27 + 8 \cdot 12$

Step 2.2.1.2

Multiply $8$ by $12$ .

$a⃗ \cdot b⃗ = 27 + 96$

Step 2.2.2

Add $27$ and $96$ .

$a⃗ \cdot b⃗ = 123$

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.

$| a⃗ | = \sqrt{3^{2} + 8^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

Raise $3$ to the power of $2$ .

$| a⃗ | = \sqrt{9 + 8^{2}}$

Step 3.2.2

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

$| a⃗ | = \sqrt{9 + 64}$

Step 3.2.3

Add $9$ and $64$ .

$| a⃗ | = \sqrt{73}$

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.

$| b⃗ | = \sqrt{9^{2} + 12^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$| b⃗ | = \sqrt{81 + 12^{2}}$

Step 4.2.2

Raise $12$ to the power of $2$ .

$| b⃗ | = \sqrt{81 + 144}$

Step 4.2.3

Add $81$ and $144$ .

$| b⃗ | = \sqrt{225}$

Step 4.2.4

Rewrite $225$ as $15^{2}$ .

$| b⃗ | = \sqrt{15^{2}}$

Step 4.2.5

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

$| b⃗ | = 15$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{123}{\sqrt{73} \cdot 15})$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $123$ and $15$ .

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Step 6.1.1

Factor $3$ out of $123$ .

$θ = \arccos (\frac{3 (41)}{\sqrt{73} \cdot 15})$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $3$ out of $\sqrt{73} \cdot 15$ .

$θ = \arccos (\frac{3 (41)}{3 (\sqrt{73} \cdot 5)})$

Step 6.1.2.2

Cancel the common factor.

$θ = \arccos (\frac{3 \cdot 41}{3 (\sqrt{73} \cdot 5)})$

Step 6.1.2.3

Rewrite the expression.

$θ = \arccos (\frac{41}{\sqrt{73} \cdot 5})$

Step 6.2

Move $5$ to the left of $\sqrt{73}$ .

$θ = \arccos (\frac{41}{5 \sqrt{73}})$

Step 6.3

Multiply $\frac{41}{5 \sqrt{73}}$ by $\frac{\sqrt{73}}{\sqrt{73}}$ .

$θ = \arccos (\frac{41}{5 \sqrt{73}} \cdot \frac{\sqrt{73}}{\sqrt{73}})$

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

Multiply $\frac{41}{5 \sqrt{73}}$ by $\frac{\sqrt{73}}{\sqrt{73}}$ .

$θ = \arccos (\frac{41 \sqrt{73}}{5 \sqrt{73} \sqrt{73}})$

Step 6.4.2

Move $\sqrt{73}$ .

$θ = \arccos (\frac{41 \sqrt{73}}{5 (\sqrt{73} \sqrt{73})})$

Step 6.4.3

Raise $\sqrt{73}$ to the power of $1$ .

$θ = \arccos (\frac{41 \sqrt{73}}{5 ({\sqrt{73}}^{1} \sqrt{73})})$

Step 6.4.4

Raise $\sqrt{73}$ to the power of $1$ .

$θ = \arccos (\frac{41 \sqrt{73}}{5 ({\sqrt{73}}^{1} {\sqrt{73}}^{1})})$

Step 6.4.5

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

$θ = \arccos (\frac{41 \sqrt{73}}{5 {\sqrt{73}}^{1 + 1}})$

Step 6.4.6

Add $1$ and $1$ .

$θ = \arccos (\frac{41 \sqrt{73}}{5 {\sqrt{73}}^{2}})$

Step 6.4.7

Rewrite ${\sqrt{73}}^{2}$ as $73$ .

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Step 6.4.7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{73}$ as $73^{\frac{1}{2}}$ .

$θ = \arccos (\frac{41 \sqrt{73}}{5 {(73^{\frac{1}{2}})}^{2}})$

Step 6.4.7.2

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

$θ = \arccos (\frac{41 \sqrt{73}}{5 \cdot 73^{\frac{1}{2} \cdot 2}})$

Step 6.4.7.3

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

$θ = \arccos (\frac{41 \sqrt{73}}{5 \cdot 73^{\frac{2}{2}}})$

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.

$θ = \arccos (\frac{41 \sqrt{73}}{5 \cdot 73^{\frac{2}{2}}})$

Step 6.4.7.4.2

Rewrite the expression.

$θ = \arccos (\frac{41 \sqrt{73}}{5 \cdot 73^{1}})$

Step 6.4.7.5

Evaluate the exponent.

$θ = \arccos (\frac{41 \sqrt{73}}{5 \cdot 73})$

Step 6.5

Multiply $5$ by $73$ .

$θ = \arccos (\frac{41 \sqrt{73}}{365})$

Step 6.6

Evaluate $\arccos (\frac{41 \sqrt{73}}{365})$ .

$θ = 16.31385242$

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