Question

Given the two planes x - y + 2z = 0 and 3x + 2y - 6z + 4 = 0. Find a parametric equation for the intersection.

Answer 1

If two planes intersect, they intersect in a line.

Setting $z=t,$ we can find the equation of the line.

$x-y+2z=0\\ \implies x-y+2t=0\\ \implies x-y=-2t\\$

$3x+2y-6z+4=0\\ \implies 3x+2y-6t+4=0\\ \implies 3x+2y=-4+6t$

$2x-2y=-4t\\ 3x+2y=-4+6t\\ 5x=-4+2t\\ x=-0.8+0.4t\\ y=x+2t=-0.8+0.4t+2t=-0.8+2.4t.$

The parametric equation for the intersection:

$x=-0.8+0.4t,y=-0.8+2.4t,z=t.$