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Chapter 1: Problem 70

Determine whether the lines \(L_{1}\) and \(L_{2}\) passing through the pairs of points are parallel, perpendicular, or neither. $$\begin{aligned} &L_{1}:(-2,-1),(1,5)\\\ &L_{2}:(1,3),(5,-5) \end{aligned}$$

Short Answer

Expert verified
The lines \(L_{1}\) and \(L_{2}\) are neither parallel nor perpendicular.

Step by step solution

01

Find the slope of \(L_{1}\)

The slope of a line passing through two points \((x_{1},y_{1})\) and \((x_{2},y_{2})\) is given by \(m=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\). For \(L_{1}\), the points are (-2,-1) and (1,5). So the slope \(m_{1}=\frac{5-(-1)}{1-(-2)}=\frac{6}{3}=2\)
02

Find the slope of \(L_{2}\)

For \(L_{2}\), the points are (1,3) and (5,-5). So, the slope \(m_{2}=\frac{-5-3}{5-1}=\frac{-8}{4}=-2\)
03

Compare the slopes

The slopes found are \(m_{1}=2\) and \(m_{2}=-2\). They are not equal, so the lines are not parallel. The product of the slopes is \(m_{1} \times m_{2}=2 \times -2=-4\), which is not -1, so the lines are not perpendicular. Therefore, the lines are neither parallel nor perpendicular.

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