Question

finding slopes of perpendicular lines
given: \\( \overleftrightarrow { p q } \perp \overleftrightarrow { p ^ { \prime } q ^ { \prime } } \\)
prove: \\( \left( m _ { \overleftrightarrow { p q } } \
ight) \left( m _ { \overleftrightarrow { p ^ { \prime } q ^ { \prime } } } \
ight) = - 1 \\)

1. \\( m _ { \overleftrightarrow { p q } } = \frac { y _ { 2 } - y _ { 1 } } { x _ { 2 } - x _ { 1 } } = \frac { \boldsymbol { \cdot } } { c - a } \\) \\( \boldsymbol { \cdot } = \\)

Explanation:

Step1: Use the slope formula

The slope formula for a line passing through two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For the line \(\overleftrightarrow{PQ}\) with points \(P(a,b)\) and \(Q(c,d)\), we substitute \(x_1 = a,y_1 = b,x_2=c,y_2 = d\) into the formula. So, \(m_{\overleftrightarrow{PQ}}=\frac{y_2 - y_1}{x_2 - x_1}=\frac{d - b}{c - a}\).

Answer:

\(\frac{d - b}{c - a}\)