Question

Consider the points below. P(−1, 3, 1), Q(0, 5, 2), R(3, 2, −1)?

(a) Find a nonzero vector orthogonal to the plane through the points P, Q, and R.


(b) Find the area of the triangle PQR.

Answer 1

Step 1:

The points on the plane are .

(a)

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

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

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

Consider .

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

Then is orthogonal to plane passing through the points .

Step 2:

(b)

Find area of the triangle .

If are the adjacent sides of the triangle then the area of the triangle is .

are the adjacent sides of the triangle then its area is .

.

The area of the triangle is .

Solution :

(a) is a non-zero vector orthogonal to plane passing through the points .

(b) The area of the triangle is .