Pre-Algebra Examples

$(1, 9)$ , $(- 3, 9)$

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.

$a⃗ \cdot b⃗ = 1 \cdot - 3 + 9 \cdot 9$

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 $- 3$ by $1$ .

$a⃗ \cdot b⃗ = - 3 + 9 \cdot 9$

Step 2.2.1.2

Multiply $9$ by $9$ .

$a⃗ \cdot b⃗ = - 3 + 81$

Step 2.2.2

Add $- 3$ and $81$ .

$a⃗ \cdot b⃗ = 78$

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.

$| a⃗ | = \sqrt{1^{2} + 9^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

One to any power is one.

$| a⃗ | = \sqrt{1 + 9^{2}}$

Step 3.2.2

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

$| a⃗ | = \sqrt{1 + 81}$

Step 3.2.3

Add $1$ and $81$ .

$| a⃗ | = \sqrt{82}$

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.

$| b⃗ | = \sqrt{{(- 3)}^{2} + 9^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$| b⃗ | = \sqrt{9 + 9^{2}}$

Step 4.2.2

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

$| b⃗ | = \sqrt{9 + 81}$

Step 4.2.3

Add $9$ and $81$ .

$| b⃗ | = \sqrt{90}$

Step 4.2.4

Rewrite $90$ as $3^{2} \cdot 10$ .

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Step 4.2.4.1

Factor $9$ out of $90$ .

$| b⃗ | = \sqrt{9 (10)}$

Step 4.2.4.2

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

$| b⃗ | = \sqrt{3^{2} \cdot 10}$

Step 4.2.5

Pull terms out from under the radical.

$| b⃗ | = 3 \sqrt{10}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{78}{\sqrt{82} (3 \sqrt{10})})$

Step 6

Simplify.

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Step 6.1

Cancel the common factor of $78$ and $3$ .

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Step 6.1.1

Factor $3$ out of $78$ .

$θ = \arccos (\frac{3 \cdot 26}{\sqrt{82} (3 \sqrt{10})})$

Step 6.1.2

Cancel the common factors.

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Step 6.1.2.1

Factor $3$ out of $\sqrt{82} (3 \sqrt{10})$ .

$θ = \arccos (\frac{3 \cdot 26}{3 (\sqrt{82} (\sqrt{10}))})$

Step 6.1.2.2

Cancel the common factor.

$θ = \arccos (\frac{3 \cdot 26}{3 (\sqrt{82} (\sqrt{10}))})$

Step 6.1.2.3

Rewrite the expression.

$θ = \arccos (\frac{26}{\sqrt{82} (\sqrt{10})})$

Step 6.2

Simplify the denominator.

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Step 6.2.1

Combine using the product rule for radicals.

$θ = \arccos (\frac{26}{\sqrt{82 \cdot 10}})$

Step 6.2.2

Multiply $82$ by $10$ .

$θ = \arccos (\frac{26}{\sqrt{820}})$

Step 6.3

Simplify the denominator.

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Step 6.3.1

Rewrite $820$ as $2^{2} \cdot 205$ .

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Step 6.3.1.1

Factor $4$ out of $820$ .

$θ = \arccos (\frac{26}{\sqrt{4 (205)}})$

Step 6.3.1.2

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

$θ = \arccos (\frac{26}{\sqrt{2^{2} \cdot 205}})$

Step 6.3.2

Pull terms out from under the radical.

$θ = \arccos (\frac{26}{2 \sqrt{205}})$

Step 6.4

Cancel the common factor of $26$ and $2$ .

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Step 6.4.1

Factor $2$ out of $26$ .

$θ = \arccos (\frac{2 \cdot 13}{2 \sqrt{205}})$

Step 6.4.2

Cancel the common factors.

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Step 6.4.2.1

Factor $2$ out of $2 \sqrt{205}$ .

$θ = \arccos (\frac{2 \cdot 13}{2 (\sqrt{205})})$

Step 6.4.2.2

Cancel the common factor.

$θ = \arccos (\frac{2 \cdot 13}{2 \sqrt{205}})$

Step 6.4.2.3

Rewrite the expression.

$θ = \arccos (\frac{13}{\sqrt{205}})$

Step 6.5

Multiply $\frac{13}{\sqrt{205}}$ by $\frac{\sqrt{205}}{\sqrt{205}}$ .

$θ = \arccos (\frac{13}{\sqrt{205}} \cdot \frac{\sqrt{205}}{\sqrt{205}})$

Step 6.6

Combine and simplify the denominator.

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Step 6.6.1

Multiply $\frac{13}{\sqrt{205}}$ by $\frac{\sqrt{205}}{\sqrt{205}}$ .

$θ = \arccos (\frac{13 \sqrt{205}}{\sqrt{205} \sqrt{205}})$

Step 6.6.2

Raise $\sqrt{205}$ to the power of $1$ .

$θ = \arccos (\frac{13 \sqrt{205}}{{\sqrt{205}}^{1} \sqrt{205}})$

Step 6.6.3

Raise $\sqrt{205}$ to the power of $1$ .

$θ = \arccos (\frac{13 \sqrt{205}}{{\sqrt{205}}^{1} {\sqrt{205}}^{1}})$

Step 6.6.4

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

$θ = \arccos (\frac{13 \sqrt{205}}{{\sqrt{205}}^{1 + 1}})$

Step 6.6.5

Add $1$ and $1$ .

$θ = \arccos (\frac{13 \sqrt{205}}{{\sqrt{205}}^{2}})$

Step 6.6.6

Rewrite ${\sqrt{205}}^{2}$ as $205$ .

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Step 6.6.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{205}$ as $205^{\frac{1}{2}}$ .

$θ = \arccos (\frac{13 \sqrt{205}}{{(205^{\frac{1}{2}})}^{2}})$

Step 6.6.6.2

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

$θ = \arccos (\frac{13 \sqrt{205}}{205^{\frac{1}{2} \cdot 2}})$

Step 6.6.6.3

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

$θ = \arccos (\frac{13 \sqrt{205}}{205^{\frac{2}{2}}})$

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.

$θ = \arccos (\frac{13 \sqrt{205}}{205^{\frac{2}{2}}})$

Step 6.6.6.4.2

Rewrite the expression.

$θ = \arccos (\frac{13 \sqrt{205}}{205^{1}})$

Step 6.6.6.5

Evaluate the exponent.

$θ = \arccos (\frac{13 \sqrt{205}}{205})$

Step 6.7

Evaluate $\arccos (\frac{13 \sqrt{205}}{205})$ .

$θ = 24.77514056$