Chapter 6 - Section 6.5 - Complex Numbers in Polar Form; DeMoivre's Theorem - Exercise Set - Page 769: 117

The slopes of the provided two lines are equal.

Work Step by Step

If the slopes of two lines are equal then the lines are parallel to each other. The slope $\left( m \right)$ of the line passing through the two points $\left( {{x}_{1}},{{x}_{2}} \right)\text{ and }\left( {{y}_{1}},{{y}_{2}} \right)$ is given by: $m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}}$. The slope of the first line passing through the point $\left( -3,-3 \right)$ and $\left( 0,3 \right)$ is: \begin{align} & {{m}_{1}}=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \\ & =\frac{3-\left( -3 \right)}{0-\left( -3 \right)} \\ & =\frac{6}{3} \\ & =2 \end{align} So, the slope of the first line is $2$ The slope of the second line passing through $\left( 0,0 \right)$ and $\left( 3,6 \right)$ is \begin{align} & {{m}_{2}}=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \\ & =\frac{0-\left( -6 \right)}{0-\left( -3 \right)} \\ & =\frac{6}{3} \\ & =2 \end{align} Thus, the slope of the second line is also $2$.

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