Pre-Algebra Examples

$(3, 3)$ , $(5, 7)$

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.

$a⃗ \cdot b⃗ = 3 \cdot 5 + 3 \cdot 7$

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 $5$ .

$a⃗ \cdot b⃗ = 15 + 3 \cdot 7$

Step 2.2.1.2

Multiply $3$ by $7$ .

$a⃗ \cdot b⃗ = 15 + 21$

Step 2.2.2

Add $15$ and $21$ .

$a⃗ \cdot b⃗ = 36$

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.

$| a⃗ | = \sqrt{3^{2} + 3^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

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

$| a⃗ | = \sqrt{9 + 3^{2}}$

Step 3.2.2

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

$| a⃗ | = \sqrt{9 + 9}$

Step 3.2.3

Add $9$ and $9$ .

$| a⃗ | = \sqrt{18}$

Step 3.2.4

Rewrite $18$ as $3^{2} \cdot 2$ .

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Step 3.2.4.1

Factor $9$ out of $18$ .

$| a⃗ | = \sqrt{9 (2)}$

Step 3.2.4.2

Rewrite $9$ as $3^{2}$ .

$| a⃗ | = \sqrt{3^{2} \cdot 2}$

Step 3.2.5

Pull terms out from under the radical.

$| a⃗ | = 3 \sqrt{2}$

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.

$| b⃗ | = \sqrt{5^{2} + 7^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$| b⃗ | = \sqrt{25 + 7^{2}}$

Step 4.2.2

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

$| b⃗ | = \sqrt{25 + 49}$

Step 4.2.3

Add $25$ and $49$ .

$| b⃗ | = \sqrt{74}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{36}{3 \sqrt{2} \sqrt{74}})$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $36$ and $3$ .

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Step 6.1.1

Factor $3$ out of $36$ .

$θ = \arccos (\frac{3 \cdot 12}{3 \sqrt{2} \sqrt{74}})$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $3$ out of $3 \sqrt{2} \sqrt{74}$ .

$θ = \arccos (\frac{3 \cdot 12}{3 (\sqrt{2} \sqrt{74})})$

Step 6.1.2.2

Cancel the common factor.

$θ = \arccos (\frac{3 \cdot 12}{3 (\sqrt{2} \sqrt{74})})$

Step 6.1.2.3

Rewrite the expression.

$θ = \arccos (\frac{12}{\sqrt{2} \sqrt{74}})$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Combine using the product rule for radicals.

$θ = \arccos (\frac{12}{\sqrt{2 \cdot 74}})$

Step 6.2.2

Multiply $2$ by $74$ .

$θ = \arccos (\frac{12}{\sqrt{148}})$

Step 6.3

Simplify the denominator.

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Step 6.3.1

Rewrite $148$ as $2^{2} \cdot 37$ .

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Step 6.3.1.1

Factor $4$ out of $148$ .

$θ = \arccos (\frac{12}{\sqrt{4 (37)}})$

Step 6.3.1.2

Rewrite $4$ as $2^{2}$ .

$θ = \arccos (\frac{12}{\sqrt{2^{2} \cdot 37}})$

Step 6.3.2

Pull terms out from under the radical.

$θ = \arccos (\frac{12}{2 \sqrt{37}})$

Step 6.4

Cancel the common factor of $12$ and $2$ .

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Step 6.4.1

Factor $2$ out of $12$ .

$θ = \arccos (\frac{2 \cdot 6}{2 \sqrt{37}})$

Step 6.4.2

Cancel the common factors.

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Step 6.4.2.1

Factor $2$ out of $2 \sqrt{37}$ .

$θ = \arccos (\frac{2 \cdot 6}{2 (\sqrt{37})})$

Step 6.4.2.2

Cancel the common factor.

$θ = \arccos (\frac{2 \cdot 6}{2 \sqrt{37}})$

Step 6.4.2.3

Rewrite the expression.

$θ = \arccos (\frac{6}{\sqrt{37}})$

Step 6.5

Multiply $\frac{6}{\sqrt{37}}$ by $\frac{\sqrt{37}}{\sqrt{37}}$ .

$θ = \arccos (\frac{6}{\sqrt{37}} \cdot \frac{\sqrt{37}}{\sqrt{37}})$

Step 6.6

Combine and simplify the denominator.

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Step 6.6.1

Multiply $\frac{6}{\sqrt{37}}$ by $\frac{\sqrt{37}}{\sqrt{37}}$ .

$θ = \arccos (\frac{6 \sqrt{37}}{\sqrt{37} \sqrt{37}})$

Step 6.6.2

Raise $\sqrt{37}$ to the power of $1$ .

$θ = \arccos (\frac{6 \sqrt{37}}{{\sqrt{37}}^{1} \sqrt{37}})$

Step 6.6.3

Raise $\sqrt{37}$ to the power of $1$ .

$θ = \arccos (\frac{6 \sqrt{37}}{{\sqrt{37}}^{1} {\sqrt{37}}^{1}})$

Step 6.6.4

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

$θ = \arccos (\frac{6 \sqrt{37}}{{\sqrt{37}}^{1 + 1}})$

Step 6.6.5

Add $1$ and $1$ .

$θ = \arccos (\frac{6 \sqrt{37}}{{\sqrt{37}}^{2}})$

Step 6.6.6

Rewrite ${\sqrt{37}}^{2}$ as $37$ .

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Step 6.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{37}$ as $37^{\frac{1}{2}}$ .

$θ = \arccos (\frac{6 \sqrt{37}}{{(37^{\frac{1}{2}})}^{2}})$

Step 6.6.6.2

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

$θ = \arccos (\frac{6 \sqrt{37}}{37^{\frac{1}{2} \cdot 2}})$

Step 6.6.6.3

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

$θ = \arccos (\frac{6 \sqrt{37}}{37^{\frac{2}{2}}})$

Step 6.6.6.4

Cancel the common factor of $2$ .

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Step 6.6.6.4.1

Cancel the common factor.

$θ = \arccos (\frac{6 \sqrt{37}}{37^{\frac{2}{2}}})$

Step 6.6.6.4.2

Rewrite the expression.

$θ = \arccos (\frac{6 \sqrt{37}}{37^{1}})$

Step 6.6.6.5

Evaluate the exponent.

$θ = \arccos (\frac{6 \sqrt{37}}{37})$

Step 6.7

Evaluate $\arccos (\frac{6 \sqrt{37}}{37})$ .

$θ = 9.4623222$