Question

Find the coordinates of the point where line through the points $A=(3,4,1)$ and $B=(5,1,6)$ crosses the $xy$-plane.

Answer 1

$A=(3,4,1);B=(5,1,6)$

The line can be written as $r=(3,4,1)+t(2,-3,5)$

This also can be written as $t=\frac{x-3}{2}=\frac{y-4}{-3}=\frac{z-1}{5}$

Since we need the coordinates of the point where the line crosses the x-y plane, we make z equal to zero.

$\therefore t=\frac{0-1}{5}=\frac{-1}{5}$

$\therefore \frac{x-3}{2}=\frac{-1}{5}$

$\Rightarrow 5x-15=-2$ or $x=\frac{13}{5}$

Similarly, $\frac{y-4}{-3}=\frac{-1}{5}$

$\therefore 5y-20=3$ or $y=\frac{23}{5}$

The coordinates thus become $(\frac{13}{5},\frac{23}{5},0)$