Basic Math Examples

$(9, - 2)$ , $(- 5, - 6)$

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.

$a⃗ \cdot b⃗ = 9 \cdot - 5 - 2 \cdot - 6$

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 $9$ by $- 5$ .

$a⃗ \cdot b⃗ = - 45 - 2 \cdot - 6$

Step 2.2.1.2

Multiply $- 2$ by $- 6$ .

$a⃗ \cdot b⃗ = - 45 + 12$

Step 2.2.2

Add $- 45$ and $12$ .

$a⃗ \cdot b⃗ = - 33$

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.

$| a⃗ | = \sqrt{9^{2} + {(- 2)}^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

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

$| a⃗ | = \sqrt{81 + {(- 2)}^{2}}$

Step 3.2.2

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

$| a⃗ | = \sqrt{81 + 4}$

Step 3.2.3

Add $81$ and $4$ .

$| a⃗ | = \sqrt{85}$

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.

$| b⃗ | = \sqrt{{(- 5)}^{2} + {(- 6)}^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$| b⃗ | = \sqrt{25 + {(- 6)}^{2}}$

Step 4.2.2

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

$| b⃗ | = \sqrt{25 + 36}$

Step 4.2.3

Add $25$ and $36$ .

$| b⃗ | = \sqrt{61}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{- 33}{\sqrt{85} \sqrt{61}})$

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.

$θ = \arccos (\frac{- 33}{\sqrt{85 \cdot 61}})$

Step 6.1.2

Multiply $85$ by $61$ .

$θ = \arccos (\frac{- 33}{\sqrt{5185}})$

Step 6.2

Move the negative in front of the fraction.

$θ = \arccos (- \frac{33}{\sqrt{5185}})$

Step 6.3

Multiply $\frac{33}{\sqrt{5185}}$ by $\frac{\sqrt{5185}}{\sqrt{5185}}$ .

$θ = \arccos (- (\frac{33}{\sqrt{5185}} \cdot \frac{\sqrt{5185}}{\sqrt{5185}}))$

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

Multiply $\frac{33}{\sqrt{5185}}$ by $\frac{\sqrt{5185}}{\sqrt{5185}}$ .

$θ = \arccos (- \frac{33 \sqrt{5185}}{\sqrt{5185} \sqrt{5185}})$

Step 6.4.2

Raise $\sqrt{5185}$ to the power of $1$ .

$θ = \arccos (- \frac{33 \sqrt{5185}}{{\sqrt{5185}}^{1} \sqrt{5185}})$

Step 6.4.3

Raise $\sqrt{5185}$ to the power of $1$ .

$θ = \arccos (- \frac{33 \sqrt{5185}}{{\sqrt{5185}}^{1} {\sqrt{5185}}^{1}})$

Step 6.4.4

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

$θ = \arccos (- \frac{33 \sqrt{5185}}{{\sqrt{5185}}^{1 + 1}})$

Step 6.4.5

Add $1$ and $1$ .

$θ = \arccos (- \frac{33 \sqrt{5185}}{{\sqrt{5185}}^{2}})$

Step 6.4.6

Rewrite ${\sqrt{5185}}^{2}$ as $5185$ .

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Step 6.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{5185}$ as $5185^{\frac{1}{2}}$ .

$θ = \arccos (- \frac{33 \sqrt{5185}}{{(5185^{\frac{1}{2}})}^{2}})$

Step 6.4.6.2

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

$θ = \arccos (- \frac{33 \sqrt{5185}}{5185^{\frac{1}{2} \cdot 2}})$

Step 6.4.6.3

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

$θ = \arccos (- \frac{33 \sqrt{5185}}{5185^{\frac{2}{2}}})$

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.

$θ = \arccos (- \frac{33 \sqrt{5185}}{5185^{\frac{2}{2}}})$

Step 6.4.6.4.2

Rewrite the expression.

$θ = \arccos (- \frac{33 \sqrt{5185}}{5185^{1}})$

Step 6.4.6.5

Evaluate the exponent.

$θ = \arccos (- \frac{33 \sqrt{5185}}{5185})$

Step 6.5

Evaluate $\arccos (- \frac{33 \sqrt{5185}}{5185})$ .

$θ = 117.27676338$