Pre-Algebra Examples

$(10, 7)$ , $(- 2, 1)$

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.

$a⃗ \cdot b⃗ = 10 \cdot - 2 + 7 \cdot 1$

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 $10$ by $- 2$ .

$a⃗ \cdot b⃗ = - 20 + 7 \cdot 1$

Step 2.2.1.2

Multiply $7$ by $1$ .

$a⃗ \cdot b⃗ = - 20 + 7$

Step 2.2.2

Add $- 20$ and $7$ .

$a⃗ \cdot b⃗ = - 13$

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.

$| a⃗ | = \sqrt{10^{2} + 7^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

Raise $10$ to the power of $2$ .

$| a⃗ | = \sqrt{100 + 7^{2}}$

Step 3.2.2

Raise $7$ to the power of $2$ .

$| a⃗ | = \sqrt{100 + 49}$

Step 3.2.3

Add $100$ and $49$ .

$| a⃗ | = \sqrt{149}$

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.

$| b⃗ | = \sqrt{{(- 2)}^{2} + 1^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$| b⃗ | = \sqrt{4 + 1^{2}}$

Step 4.2.2

One to any power is one.

$| b⃗ | = \sqrt{4 + 1}$

Step 4.2.3

Add $4$ and $1$ .

$| b⃗ | = \sqrt{5}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{- 13}{\sqrt{149} \sqrt{5}})$

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.

$θ = \arccos (\frac{- 13}{\sqrt{149 \cdot 5}})$

Step 6.1.2

Multiply $149$ by $5$ .

$θ = \arccos (\frac{- 13}{\sqrt{745}})$

Step 6.2

Move the negative in front of the fraction.

$θ = \arccos (- \frac{13}{\sqrt{745}})$

Step 6.3

Multiply $\frac{13}{\sqrt{745}}$ by $\frac{\sqrt{745}}{\sqrt{745}}$ .

$θ = \arccos (- (\frac{13}{\sqrt{745}} \cdot \frac{\sqrt{745}}{\sqrt{745}}))$

Step 6.4

Combine and simplify the denominator.

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Step 6.4.1

Multiply $\frac{13}{\sqrt{745}}$ by $\frac{\sqrt{745}}{\sqrt{745}}$ .

$θ = \arccos (- \frac{13 \sqrt{745}}{\sqrt{745} \sqrt{745}})$

Step 6.4.2

Raise $\sqrt{745}$ to the power of $1$ .

$θ = \arccos (- \frac{13 \sqrt{745}}{{\sqrt{745}}^{1} \sqrt{745}})$

Step 6.4.3

Raise $\sqrt{745}$ to the power of $1$ .

$θ = \arccos (- \frac{13 \sqrt{745}}{{\sqrt{745}}^{1} {\sqrt{745}}^{1}})$

Step 6.4.4

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

$θ = \arccos (- \frac{13 \sqrt{745}}{{\sqrt{745}}^{1 + 1}})$

Step 6.4.5

Add $1$ and $1$ .

$θ = \arccos (- \frac{13 \sqrt{745}}{{\sqrt{745}}^{2}})$

Step 6.4.6

Rewrite ${\sqrt{745}}^{2}$ as $745$ .

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Step 6.4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{745}$ as $745^{\frac{1}{2}}$ .

$θ = \arccos (- \frac{13 \sqrt{745}}{{(745^{\frac{1}{2}})}^{2}})$

Step 6.4.6.2

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

$θ = \arccos (- \frac{13 \sqrt{745}}{745^{\frac{1}{2} \cdot 2}})$

Step 6.4.6.3

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

$θ = \arccos (- \frac{13 \sqrt{745}}{745^{\frac{2}{2}}})$

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.

$θ = \arccos (- \frac{13 \sqrt{745}}{745^{\frac{2}{2}}})$

Step 6.4.6.4.2

Rewrite the expression.

$θ = \arccos (- \frac{13 \sqrt{745}}{745^{1}})$

Step 6.4.6.5

Evaluate the exponent.

$θ = \arccos (- \frac{13 \sqrt{745}}{745})$

Step 6.5

Evaluate $\arccos (- \frac{13 \sqrt{745}}{745})$ .

$θ = 118.44292862$