Trigonometry Examples

$(\frac{1}{6}, - \frac{1}{3})$ , $(\frac{5}{6}, 3)$

Step 1

To find the position vector, subtract the initial point vector $P$ from the terminal point vector $Q$ .

$Q - P = (\frac{5}{6 i} + 3 j) - (\frac{1}{6 i} - \frac{1}{3 j})$

Step 2

Simplify each term.

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Step 2.1

Multiply the numerator and denominator of $\frac{5}{6 i}$ by the conjugate of $6 i$ to make the denominator real.

$\frac{5}{6 i} \cdot \frac{i}{i} + 3 j - (\frac{1}{6 i} - \frac{1}{3 j})$

Step 2.2

Multiply.

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Step 2.2.1

Combine.

$\frac{5 i}{6 i i} + 3 j - (\frac{1}{6 i} - \frac{1}{3 j})$

Step 2.2.2

Simplify the denominator.

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Step 2.2.2.1

Add parentheses.

$\frac{5 i}{6 (i i)} + 3 j - (\frac{1}{6 i} - \frac{1}{3 j})$

Step 2.2.2.2

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

$\frac{5 i}{6 (i^{1} i)} + 3 j - (\frac{1}{6 i} - \frac{1}{3 j})$

Step 2.2.2.3

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

$\frac{5 i}{6 (i^{1} i^{1})} + 3 j - (\frac{1}{6 i} - \frac{1}{3 j})$

Step 2.2.2.4

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

$\frac{5 i}{6 i^{1 + 1}} + 3 j - (\frac{1}{6 i} - \frac{1}{3 j})$

Step 2.2.2.5

Add $1$ and $1$ .

$\frac{5 i}{6 i^{2}} + 3 j - (\frac{1}{6 i} - \frac{1}{3 j})$

Step 2.2.2.6

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

$\frac{5 i}{6 \cdot - 1} + 3 j - (\frac{1}{6 i} - \frac{1}{3 j})$

Step 2.3

Multiply $6$ by $- 1$ .

$\frac{5 i}{- 6} + 3 j - (\frac{1}{6 i} - \frac{1}{3 j})$

Step 2.4

Move the negative in front of the fraction.

$- \frac{5 i}{6} + 3 j - (\frac{1}{6 i} - \frac{1}{3 j})$

Step 2.5

Simplify each term.

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Step 2.5.1

Multiply the numerator and denominator of $\frac{1}{6 i}$ by the conjugate of $6 i$ to make the denominator real.

$- \frac{5 i}{6} + 3 j - (\frac{1}{6 i} \cdot \frac{i}{i} - \frac{1}{3 j})$

Step 2.5.2

Multiply.

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Step 2.5.2.1

Combine.

$- \frac{5 i}{6} + 3 j - (\frac{1 i}{6 i i} - \frac{1}{3 j})$

Step 2.5.2.2

Multiply $i$ by $1$ .

$- \frac{5 i}{6} + 3 j - (\frac{i}{6 i i} - \frac{1}{3 j})$

Step 2.5.2.3

Simplify the denominator.

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Step 2.5.2.3.1

Add parentheses.

$- \frac{5 i}{6} + 3 j - (\frac{i}{6 (i i)} - \frac{1}{3 j})$

Step 2.5.2.3.2

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

$- \frac{5 i}{6} + 3 j - (\frac{i}{6 (i^{1} i)} - \frac{1}{3 j})$

Step 2.5.2.3.3

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

$- \frac{5 i}{6} + 3 j - (\frac{i}{6 (i^{1} i^{1})} - \frac{1}{3 j})$

Step 2.5.2.3.4

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

$- \frac{5 i}{6} + 3 j - (\frac{i}{6 i^{1 + 1}} - \frac{1}{3 j})$

Step 2.5.2.3.5

Add $1$ and $1$ .

$- \frac{5 i}{6} + 3 j - (\frac{i}{6 i^{2}} - \frac{1}{3 j})$

Step 2.5.2.3.6

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

$- \frac{5 i}{6} + 3 j - (\frac{i}{6 \cdot - 1} - \frac{1}{3 j})$

Step 2.5.3

Multiply $6$ by $- 1$ .

$- \frac{5 i}{6} + 3 j - (\frac{i}{- 6} - \frac{1}{3 j})$

Step 2.5.4

Move the negative in front of the fraction.

$- \frac{5 i}{6} + 3 j - (- \frac{i}{6} - \frac{1}{3 j})$

Step 2.6

Apply the distributive property.

$- \frac{5 i}{6} + 3 j - - \frac{i}{6} - - \frac{1}{3 j}$

Step 2.7

Multiply $- - \frac{i}{6}$ .

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Step 2.7.1

Multiply $- 1$ by $- 1$ .

$- \frac{5 i}{6} + 3 j + 1 \frac{i}{6} - - \frac{1}{3 j}$

Step 2.7.2

Multiply $\frac{i}{6}$ by $1$ .

$- \frac{5 i}{6} + 3 j + \frac{i}{6} - - \frac{1}{3 j}$

Step 2.8

Multiply $- - \frac{1}{3 j}$ .

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Step 2.8.1

Multiply $- 1$ by $- 1$ .

$- \frac{5 i}{6} + 3 j + \frac{i}{6} + 1 \frac{1}{3 j}$

Step 2.8.2

Multiply $\frac{1}{3 j}$ by $1$ .

$- \frac{5 i}{6} + 3 j + \frac{i}{6} + \frac{1}{3 j}$

Step 3

Simplify terms.

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Step 3.1

Combine the numerators over the common denominator.

$3 j + \frac{- 5 i + i}{6} + \frac{1}{3 j}$

Step 3.2

Add $- 5 i$ and $i$ .

$3 j + \frac{- 4 i}{6} + \frac{1}{3 j}$

Step 4

Simplify each term.

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Step 4.1

Cancel the common factor of $- 4$ and $6$ .

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Step 4.1.1

Factor $2$ out of $- 4 i$ .

$3 j + \frac{2 (- 2 i)}{6} + \frac{1}{3 j}$

Step 4.1.2

Cancel the common factors.

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Step 4.1.2.1

Factor $2$ out of $6$ .

$3 j + \frac{2 (- 2 i)}{2 (3)} + \frac{1}{3 j}$

Step 4.1.2.2

Cancel the common factor.

$3 j + \frac{2 (- 2 i)}{2 \cdot 3} + \frac{1}{3 j}$

Step 4.1.2.3

Rewrite the expression.

$3 j + \frac{- 2 i}{3} + \frac{1}{3 j}$

Step 4.2

Move the negative in front of the fraction.

$3 j - \frac{2 i}{3} + \frac{1}{3 j}$

Step 5