Help with simple cross product problems?

Question

1. Let A = (-1, 3, 0), B = (4, 3, 4), and C = (1, -1, 5). Find the exact area of the triangle determined by these three points.

2. Find a unit vector perpendicular to i + j and i - j - k. Give an exact answer.

Any help with these would be greatly appreciated. Thanks!

Answer 1

2.  The cross product of these is a vector perpendicular to both, so we first find the cross product: <1,1,0> X <1,-1,-1> = <-1,1,-2>.  But this is not a unit vector, so we find its length, which is

sqrt((-1)^2 + 1^2 + (-2)^2) = sqrt(6).  The unit vector we want is

(1/sqrt(6))<-1,1,-2>.