Show that the lines

Question:

Show that the lines $\frac{x-5}{7}=\frac{y+2}{-5}=\frac{z}{1}$ and $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$ are perpendicular to each other.

Solution:

The equations of the given lines are $\frac{x-5}{7}=\frac{y+2}{-5}=\frac{z}{1}$ and $\frac{x}{1}=\frac{y}{2}=\frac{z}{3}$

The direction ratios of the given lines are 7, −5, 1 and 1, 2, 3 respectively.

Two lines with direction ratios, $a_{1}, b_{1}, c_{1}$ and $a_{2}, b_{2}, c_{2}$, are perpendicular to each other, if $a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0$

$\therefore 7 \times 1+(-5) \times 2+1 \times 3$

$=7-10+3$

$=0$

Therefore, the given lines are perpendicular to each other.