How do you find an equation for the plane that contains the line with parametric equations l=(8 - 7t, -5 - 2t , 5 - t) and is parallel to the line with parametric equations x=3 + t, y=-7 + 9t, z=8 - 6t?

1 Answer
Sep 30, 2016

Pi->21 x - 43 y - 61 z-78=0

Explanation:

Given two lines L_1,L_2 determine a plane Pi such that

L_1 in Pi and L_2 !in Pi

Here

L_1 -> p = p_1 + lambda vec v_1
L_2 -> p = p_2 + lambda vec v_2

with

p_1 = (8,-5,5), vec v_1 = (-7,-2,-1)
p_2 = (3,-7,8), vec v_2 = (1,9,-6)

The plane equation is

Pi->p = p_1+lambda_1 vec v_1+lambda_2 vec v_2 because

L_1 in Pi ( make lambda_2=0 ) and is parallel to L_2 ( making lambda_2=0)

The nonparametric plane equation can be easily obtained remembering that for the plane

<< p-p_1,vec n >> = 0

here p = (x,y,z) and vec n = vec v_1 xx vec v_2 = (21,-43,-61) so

Pi->21 x - 43 y - 61 z-78=0