Final answer:
The unit vector with a positive first coordinate that is orthogonal to the plane through points P, Q, and R is (1, 0, 0) after calculating the cross product of vectors in the plane and normalizing it.
Step-by-step explanation:
To find a unit vector that is orthogonal to a plane defined by three points P, Q, and R, we need to first find the normal vector to the plane. This can be done by calculating the cross product of two vectors that lie in the plane. We can use the vectors ▲PQ and ▲PR for this purpose. The calculations are as follows:
Let ▲PQ = Q - P = (0 - (-4), 3 - (-1), 3 - (-1)) = (4, 4, 4),
and let ▲PR = R - P = (0 - (-4), 3 - (-1), 5 - (-1)) = (4, 4, 6).
The cross product of ▲PQ and ▲PR is:
▲PQ x ▲PR = | i j k|
| 4 4 4|
| 4 4 6| = (4*6 - 4*4)i - (4*4 - 4*4)j + (4*4 - 4*4)k = (8, 0, 0).
Since we have a normal vector (8, 0, 0), we see that the i component is already positive, and the vector also has a magnitude of 8. To find the unit vector, we divide by the magnitude:The unit vector is (8/8, 0/8, 0/8) = (1, 0, 0), which has a positive first coordinate and is orthogonal to the plane.