Find a unit vector with positive first coordinate that is orthogonal to the plane through the points P = (-4, 3, -2), Q = (1, 8, 3), and R = (1, 8, 5).

Sure, here are the steps to find the unit vector:

Step 1: Find two vectors that are in the plane. We can get these by subtracting the coordinates of the points:

Vector PQ = Q - P = (1 - -4, 8 - 3, 3 - -2) = (5, 5, 5) Vector PR = R - P = (1 - -4, 8 - 3, 5 - -2) = (5, 5, 7)

Step 2: Find a vector that is orthogonal (perpendicular) to the plane. We can get this by taking the cross product of the two vectors we found in step 1:

PQ x PR = (57 - 55, 55 - 57, 55 - 55) = (10, -10, 0)

Step 3: Normalize the orthogonal vector to get a unit vector. We do this by dividing each component of the vector by its magnitude:

Magnitude = sqrt(10^2 + -10^2 + 0^2) = sqrt(200)

Unit vector = (10/sqrt(200), -10/sqrt(200), 0/sqrt(200)) = (1/sqrt(2), -1/sqrt(2), 0)

However, we want a unit vector with a positive first coordinate. The vector we found has a negative first coordinate, so we multiply it by -1 to get the final answer:

Final unit vector = (-1/sqrt(2), 1/sqrt(2), 0)