Find a unit vector with positive first coordinate that is orthogonal to the plane?

Question

Find a unit vector with positive first coordinate that is orthogonal to the plane through the points
P = (-4, 5, 4), Q = (-1, 8, 7), and R = (-1, 8, 8)
Thanx guys. I have trouble solving it.

Find a unit vector with positive first coordinate that is orthogonal to the plane through the points
P = (-4, 5, 4)
Q = (-1, 8, 7)
R = (-1, 8, 8)
Thanx guys. I have trouble solving it.

Answer 1

First find two directional vectors for the plane from the three given points P = (-4, 5, 4), Q = (-1, 8, 7), and R = (-1, 8, 8).

PQ = = <-1+4, 8-5, 7-4> = <3, 3, 3>
PR = = <-1+4, 8-5, 8-4> = <3, 3, 4>

To find a vector n, orthogonal to both of them take the cross product.

n = PQ X PR = <3, 3, 3> X <3, 3, 4> = <3, -3, 0>

To find a unit vector, divide by the n by its magnitude.

|| n || = √[3² + (-3)² + 0²] = √(9 + 9 + 0) = √18 = 3√2


The desired unit vector is:

n / || n || = <3, -3, 0> / (3√2) = <1/√2, -1/√2, 0>