Question

Consider the planes given by the equations
2y−2x−z=2
x−2y+3z=7
(a) Find a vector v parallel to the line of intersection of the planes.

(b)Find the equation of a plane through the origin which is perpendicular to the line of intersection of these two planes.

For part a) I just used the cross product of the vectors and got -8i-7j-2k
Then for part b) I used the vector I got for part a) and the point (0,0,0) to get -8x-7y-2z=0

Both answers are wrong and I don't know why.

Answer 1

Your method is correct, except you must have made an arithmetic error,
I got (-4, -5, -2) as the cross-product

a) which we can change to (4,5,2)
b) since we have a point (0,0,0)
the equation of the required plane is
4x + 5y + 2z = 0

Another way:

Add the two equations
-x + 2z = 9
x = 2z-9

let z = 0, ----> x = -9
sub into 2nd equation, y = -8

let z = 4, ---> x = -1
sub into 2nd equation, y = 2
So we have 2 points on the line of intersection,
(-9, -8, 0) and (-1, 2,4)
and a direction vector would be
(8, 10, 4)
or reduced to (4,5,2) as above