The plane P1 has equation r = (2,1,1)+lambda(-2,1,8)+mu(1,-3,-9)

Question

The plane P2 has equation r = (2,0,1)+s(1,2,1)+t(1,1,1)

Show that for points that lie on P1 and P2, lambda=mu

I have no idea how to start... help! Thanks!

Answer 1

I will use l for lambda and m for mu

Plane1: (x,y,z) = (2,1,1) + l(-2,1,8) + m(1,-3,-9)
Plane2: (x,y,z) = (2,0,1) + s(1,2,1) + t(1,1,1)

You want the x's, the y's and the z's to be the same

2 - 2l + m = 2 + s + t
s + t + 2l - m = 0 , #1

1 + l - 3m = 0 + 2s + t
2s + t - l + 3m = 1 , #2

1 + 8l - 9m = 1 + s + t
s+ t - 8l + 9m = 0 , #3

#1 - #3:
10l - 10m = 0
l = m
lamda = mu

Now that wasn't bad, perhaps a bit lucky the way it worked out