Question

a vector parallel to the line of intersection of the planes. x-2y-z=6 and 3x-y+z=4

the answer is <3,4,-5>

so first, I took the cross product of <1,-2,-1> and <3,-1,1> which I think is v. in the equation p0 + tv. << this is the equation for the line of intersection I am pretty sure. But, I'm not sure what to do after that.

Answer 1

+i +j +k
+1 -2 -1
+3 -1 +1
================== cross product
(-3)i + (-4)j + (+5)k

so yes V = -3 i - 4 j + 5 k
opposite to that is also parallel
+3 i + 4 j -5 k

add any old point to that and it is still parallel

Answer 2

another way: find the intersection line of the two planes

add the two equations,
4x -3y = 10
y = (4x-10)/3

pick any "nice" value of x

let x = 1, y = -2
in the 2nd equation:
3 + 2 + z = 4 , z = --1 --> point (1,-2,-1)

let x = 10, y = 10,
in the 2nd
30 - 10 + z = 4, z = -16 -->point(10, 10, -16)

vector = <9,12,-15> or reduced to <3, 4, -5>