Show that the four points A, B, C and D with position vectors and respectively are coplanar if and only if

Given that,

⇒

Let A, B, C and D be coplanar.

As we know that, the vectors , , and will be coplanar if there exist scalar x, y, z, u not all zero simultaneously such that .

Then, we can write

Where, (x + y + z + u) = 0

Provided x, y, z, u are scalars not all simultaneously zero.

Let x = 3, y = –2, z = 1 and u = –2

So, we get

Thus, A, B, C and D are coplanar if .

⇐

If is true.

Rearranging it, we get

Dividing this from the sum of its coefficient (that is, 4) on both sides,

Or

⇒ There is a point say P, which divides the line AC in ratio 1:3 and BD in ratio 2:2 internally.

Thus, P is the point of interaction of AC and BD.

As, vectors parallel to the same plane, or lie on the same plane are called coplanar vectors.

Hence, A, B, C and D are coplanar.