Hi there I was attempting a previous exam question on vector products but I was not sure it I had done it correctly. Could someone please look over my solution and help me out to part B.

Show that the line L: $\displaystyle \frac{x-2}{1} = \frac{y+3}{-2}=\frac{z-1}{1}$

lies on the Plane P, $\displaystyle 3x + y - z = 2 $ and find:

(a) the equation of the plane that contains L and is perpendicular to P.

(b) equations for the line P that is perpendicular to L and intersects it in (2,-3,1)

Solution:

To show that the line lies on the plane:

I got the parametric form of L, giving x = t + 2, y = -2t -3 and z = t + 1

I then put this in the equation of the plane which gave me 2, so I said since LHS = RHS the line lies on the plane.

for part (a):

I did the vector product of the normal vector x direction vector of L:

which gave me (-1,-4,-3) and then I got the resulting plane equation as

x + 4y +3z = 17. Is this correct?

Thanks.