Question

Show that the points (0, – 1, 0), (2, 1, – 1), (1, 1, 1), (3, 3, 0) are coplanar.

Answer 1

Let A ≡ (0, – 1, 0), B ≡ (2, 1, – 1), C ≡ (1, 1, 1) and D ≡ (3, 3, 0) 

Equation of a plane through A (0, – 1, 0) is 

a (x – 0) + b (y + 1) + c (z – 0) = 0 

or, ax + by + cz + b = 0 ..... (1) 

If plane (1) passes through B (2, 1, – 1) and C (1, 1, 1) 

Then 2a + 2b – c = 0 ..... (2) 

and a + 2b + c = 0 ..... (3) 

From (2) and (3), we have

Putting the value of a, b, c, in (1), equation of required plane is 

4kx – 3k(y + 1) + 2kz = 0 

or, 4x – 3y + 2z – 3 = 0 ..... (2) 

Clearly point D (3, 3, 0) lies on plane (2) 

Thus points D lies on the plane passing through A, B, C and hence points A, B, C and D are coplanar.