Question

(a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R, and (b) find the area of triangle PQR.

Answer 1

Step 1:

(a) The points on the plane are and

The points are lies on the plane then their vectors are lie on the same plane.

If are the two points then the component form of vector is

If and are the two points then the component form  of vector  is

Consider .

From geometric properties of the cross product, is perpendicular to both .

Thus  is perpendicular to plane passing through the points .

Step 2:

Area of the is half of the area of the parallelogram with adjacent sides .

Area of the parallelogram with adjacent sides is length of the cross product of .

Area of the parallelogram is

Area of the is .

Solution:

is a non-zero vector perpendicular to plane passing through the points and .

Area of the is .