How to find a non-zero vector which is orthogonal to the plane through the given points P, Q and R ?

Question:

Consider the points here, P(1,0,1), Q(-2,1,3), R(4,2,5).

(a) Find a non-zero vector which is orthogonal to the plane through the given points P, Q and R.

(b) Find the area of the triangle PQR.

Answer:

(a)

To find the orthogonal vector to the plane, we need to know two vectors on that plane. Then take the cross product of those vectors and it will be the orthogonal vector to the plane. If cross product be zero, try the cross product with different vectors on the plane.

Figure 1

First find the position vectors of the points.

Let p, q and r be the position vectors of P,Q and R respectively. ( Using cartesian coordinates)

p=1i+0j+1k , q=(-2)i+1j+3k , r =4i+2j+5k

Then find two vectors on the plane using position vectors of points.

Therefore <0,18,-9> is a orthogonal vector to the plane.

(b)

Figure 2

Area of a parallelogram is the magnitude of the cross product of two adjacent vectors.

Area of PQR triangle is equal to the half of PQRS parallelogram.

Thank you for reading

By Thaveesha Handapangoda. For videos click here