Pre-Algebra Examples

$(1, 2.22)$ , $(2, 3.32)$

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.

$a⃗ \cdot b⃗ = 1 \cdot 2 + 2.22 \cdot 3.32$

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 $2$ by $1$ .

$a⃗ \cdot b⃗ = 2 + 2.22 \cdot 3.32$

Step 2.2.1.2

Multiply $2.22$ by $3.32$ .

$a⃗ \cdot b⃗ = 2 + 7.3704$

Step 2.2.2

Add $2$ and $7.3704$ .

$a⃗ \cdot b⃗ = 9.3704$

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.

$| a⃗ | = \sqrt{1^{2} + {2.22}^{2}}$

Step 3.2

Simplify.

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Step 3.2.1

One to any power is one.

$| a⃗ | = \sqrt{1 + {2.22}^{2}}$

Step 3.2.2

Raise $2.22$ to the power of $2$ .

$| a⃗ | = \sqrt{1 + 4.9284}$

Step 3.2.3

Add $1$ and $4.9284$ .

$| a⃗ | = \sqrt{5.9284}$

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.

$| b⃗ | = \sqrt{2^{2} + {3.32}^{2}}$

Step 4.2

Simplify.

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Step 4.2.1

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

$| b⃗ | = \sqrt{4 + {3.32}^{2}}$

Step 4.2.2

Raise $3.32$ to the power of $2$ .

$| b⃗ | = \sqrt{4 + 11.0224}$

Step 4.2.3

Add $4$ and $11.0224$ .

$| b⃗ | = \sqrt{15.0224}$

Step 5

Substitute the values into the formula.

$θ = \arccos (\frac{9.3704}{\sqrt{5.9284} \sqrt{15.0224}})$

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.

$θ = \arccos (\frac{9.3704}{\sqrt{5.9284 \cdot 15.0224}})$

Step 6.1.2

Multiply $5.9284$ by $15.0224$ .

$θ = \arccos (\frac{9.3704}{\sqrt{89.05879616}})$

Step 6.2

Multiply $\frac{9.3704}{\sqrt{89.05879616}}$ by $\frac{\sqrt{89.05879616}}{\sqrt{89.05879616}}$ .

$θ = \arccos (\frac{9.3704}{\sqrt{89.05879616}} \cdot \frac{\sqrt{89.05879616}}{\sqrt{89.05879616}})$

Step 6.3

Combine and simplify the denominator.

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Step 6.3.1

Multiply $\frac{9.3704}{\sqrt{89.05879616}}$ by $\frac{\sqrt{89.05879616}}{\sqrt{89.05879616}}$ .

$θ = \arccos (\frac{9.3704 \sqrt{89.05879616}}{\sqrt{89.05879616} \sqrt{89.05879616}})$

Step 6.3.2

Raise $\sqrt{89.05879616}$ to the power of $1$ .

$θ = \arccos (\frac{9.3704 \sqrt{89.05879616}}{{\sqrt{89.05879616}}^{1} \sqrt{89.05879616}})$

Step 6.3.3

Raise $\sqrt{89.05879616}$ to the power of $1$ .

$θ = \arccos (\frac{9.3704 \sqrt{89.05879616}}{{\sqrt{89.05879616}}^{1} {\sqrt{89.05879616}}^{1}})$

Step 6.3.4

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

$θ = \arccos (\frac{9.3704 \sqrt{89.05879616}}{{\sqrt{89.05879616}}^{1 + 1}})$

Step 6.3.5

Add $1$ and $1$ .

$θ = \arccos (\frac{9.3704 \sqrt{89.05879616}}{{\sqrt{89.05879616}}^{2}})$

Step 6.3.6

Rewrite ${\sqrt{89.05879616}}^{2}$ as $89.05879616$ .

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Step 6.3.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{89.05879616}$ as ${89.05879616}^{\frac{1}{2}}$ .

$θ = \arccos (\frac{9.3704 \sqrt{89.05879616}}{{({89.05879616}^{\frac{1}{2}})}^{2}})$

Step 6.3.6.2

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

$θ = \arccos (\frac{9.3704 \sqrt{89.05879616}}{{89.05879616}^{\frac{1}{2} \cdot 2}})$

Step 6.3.6.3

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

$θ = \arccos (\frac{9.3704 \sqrt{89.05879616}}{{89.05879616}^{\frac{2}{2}}})$

Step 6.3.6.4

Cancel the common factor of $2$ .

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Step 6.3.6.4.1

Cancel the common factor.

$θ = \arccos (\frac{9.3704 \sqrt{89.05879616}}{{89.05879616}^{\frac{2}{2}}})$

Step 6.3.6.4.2

Rewrite the expression.

$θ = \arccos (\frac{9.3704 \sqrt{89.05879616}}{{89.05879616}^{1}})$

Step 6.3.6.5

Evaluate the exponent.

$θ = \arccos (\frac{9.3704 \sqrt{89.05879616}}{89.05879616})$

Step 6.4

Multiply $9.3704$ by $\sqrt{89.05879616}$ .

$θ = \arccos (\frac{88.42937192}{89.05879616})$

Step 6.5

Divide $88.42937192$ by $89.05879616$ .

$θ = \arccos (0.99293248)$

Step 6.6

Evaluate $\arccos (0.99293248)$ .

$θ = 6.81596047$