Question

Find the coordinates of the point where the line through the point $(3,-4,-5)$ and $(2,-3,1)$ crosses the plane $2x+y+z=7.$

Answer 1

The parametric representation of the line can be written as $r=(3,-4,-5)+t\left[\right(3-2),(-4-(-3)),(-5-1\left)\right]=(3,-4,-5)+t(1,-1,-6)$

Thus, a point on the line would have coordinates as under:

$x=3+t,y=-4-t,z=-5-6t$

Since this point has to also lie on the plane, we substitute the coordinates into the plane equation and obtain

$2(3+t)+(-4-t)+(-5-6t)=7$

$\Rightarrow 6+2t-4-t-5-6t=7$

$\Rightarrow -3-5t=7$

$\Rightarrow 5t=-10$ or $t=-2$

Re-substituting t in the coordinates, we get $x=3-2=1,y=-4-t=-2,z=-5+12=7$

The point is thus $(1,-2,7)$