Find a unit vector with positive first coordinate?

Question

that is orthogonal to the plane through the points P = (3, -3, 0), Q = (5, -1, 2), and R = (5, -1, 6).

Answer 1

Here's how to do it:

Find PQ and PR. [PQ = Q - P ; PR = R - P]

Now, form their cross product (PQ x PR). This will be orthogonal to the plane.

Normalize it. (Divide it by its magnitude.)

You will now have a unit vector orthogonal to the plane.

If the first coordinate isn't positive, simply flip the signs of all three components (negate the vector).

Answer 2

Find the equation of the plane through the three given points, and put it into the form ax + by + cz + d = 0. The vector ai + bj + ck is orthogonal to the plane, so all you have to do is normalize it.