Finite Math Examples

$v \cdot w = (- 5 i + 6 j) (- 2 i - 3 i)$

Step 1

Rewrite the equation as $(- 5 i + 6 j) (- 2 i - 3 i) = v \cdot w$ .

$(- 5 i + 6 j) (- 2 i - 3 i) = v \cdot w$

Step 2

Subtract $3 i$ from $- 2 i$ .

$(- 5 i + 6 j) (- 5 i) = v \cdot w$

Step 3

Divide each term in $(- 5 i + 6 j) (- 5 i) = v \cdot w$ by $- 5 i$ and simplify.

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Step 3.1

Divide each term in $(- 5 i + 6 j) (- 5 i) = v \cdot w$ by $- 5 i$ .

$\frac{(- 5 i + 6 j) (- 5 i)}{- 5 i} = \frac{v \cdot w}{- 5 i}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $- 5$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{(- 5 i + 6 j) (- 5 i)}{- 5 i} = \frac{v \cdot w}{- 5 i}$

Step 3.2.1.2

Rewrite the expression.

$\frac{(- 5 i + 6 j) (i)}{i} = \frac{v \cdot w}{- 5 i}$

Step 3.2.2

Cancel the common factor of $i$ .

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Step 3.2.2.1

Cancel the common factor.

$\frac{(- 5 i + 6 j) i}{i} = \frac{v \cdot w}{- 5 i}$

Step 3.2.2.2

Divide $- 5 i + 6 j$ by $1$ .

$- 5 i + 6 j = \frac{v \cdot w}{- 5 i}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Multiply the numerator and denominator of $\frac{v \cdot w}{- 5 i}$ by the conjugate of $- 5 i$ to make the denominator real.

$- 5 i + 6 j = \frac{v \cdot w}{- 5 i} \cdot \frac{i}{i}$

Step 3.3.2

Multiply.

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Step 3.3.2.1

Combine.

$- 5 i + 6 j = \frac{v \cdot w i}{- 5 i i}$

Step 3.3.2.2

Simplify the denominator.

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Step 3.3.2.2.1

Add parentheses.

$- 5 i + 6 j = \frac{v w i}{- 5 (i i)}$

Step 3.3.2.2.2

Raise $i$ to the power of $1$ .

$- 5 i + 6 j = \frac{v w i}{- 5 (i^{1} i)}$

Step 3.3.2.2.3

Raise $i$ to the power of $1$ .

$- 5 i + 6 j = \frac{v w i}{- 5 (i^{1} i^{1})}$

Step 3.3.2.2.4

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

$- 5 i + 6 j = \frac{v w i}{- 5 i^{1 + 1}}$

Step 3.3.2.2.5

Add $1$ and $1$ .

$- 5 i + 6 j = \frac{v w i}{- 5 i^{2}}$

Step 3.3.2.2.6

Rewrite $i^{2}$ as $- 1$ .

$- 5 i + 6 j = \frac{v w i}{- 5 \cdot - 1}$

Step 3.3.3

Multiply $- 5$ by $- 1$ .

$- 5 i + 6 j = \frac{v w i}{5}$

Step 4

Add $5 i$ to both sides of the equation.

$6 j = \frac{v w i}{5} + 5 i$

Step 5

Divide each term in $6 j = \frac{v w i}{5} + 5 i$ by $6$ and simplify.

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Step 5.1

Divide each term in $6 j = \frac{v w i}{5} + 5 i$ by $6$ .

$\frac{6 j}{6} = \frac{\frac{v w i}{5}}{6} + \frac{5 i}{6}$

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $6$ .

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Step 5.2.1.1

Cancel the common factor.

$\frac{6 j}{6} = \frac{\frac{v w i}{5}}{6} + \frac{5 i}{6}$

Step 5.2.1.2

Divide $j$ by $1$ .

$j = \frac{\frac{v w i}{5}}{6} + \frac{5 i}{6}$

Step 5.3

Simplify the right side.

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Step 5.3.1

Simplify each term.

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Step 5.3.1.1

Multiply the numerator by the reciprocal of the denominator.

$j = \frac{v w i}{5} \cdot \frac{1}{6} + \frac{5 i}{6}$

Step 5.3.1.2

Multiply $\frac{v w i}{5} \cdot \frac{1}{6}$ .

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Step 5.3.1.2.1

Multiply $\frac{v w i}{5}$ by $\frac{1}{6}$ .

$j = \frac{v w i}{5 \cdot 6} + \frac{5 i}{6}$

Step 5.3.1.2.2

Multiply $5$ by $6$ .

$j = \frac{v w i}{30} + \frac{5 i}{6}$