Final answer:
To find a unit vector that is orthogonal to the plane through the points P, Q, and R, we can first find two vectors that lie in the plane and then take their cross product.
Step-by-step explanation:
To find a unit vector that is orthogonal to the plane through the points P, Q, and R, we can first find two vectors that lie in the plane and then take their cross product. Let's call the vectors formed by the points PQ and PR. The vector PQ is given by (1 - (-1), 1 - (-1), -2 - (-4)) = (2, 2, 2). The vector PR is given by (1 - (-1), 1 - (-1), -1 - (-4)) = (2, 2, 3).
Now, we can take the cross product of these two vectors to get a vector that is normal (orthogonal) to the plane: (2, 2, 2) x (2, 2, 3) = (-2, 2, 0).
Finally, we can find the unit vector in the direction of (-2, 2, 0) by dividing it by its magnitude: ||(-2, 2, 0)|| = sqrt((-2)^2 + 2^2 + 0^2) = sqrt(8) = 2 sqrt(2). So, the unit vector is (-2/sqrt(8), 2/sqrt(8), 0) = (-sqrt(2)/2, sqrt(2)/2, 0).