Prove that a necessary and sufficient condition for three vectors and to be coplanar is that there exist scalars not all zero simultaneously such that

Given: The vectors , and .

To Prove: (a). Necessary condition: The vectors , and will be coplanar if there exist scalar l, m, n not all zero simultaneously such that .

(b). Sufficient condition: For vectors , and , there exist scalar l, m, n not all zero simultaneously such that

Proof:

(a). Necessary condition: Let , and are three coplanar vectors.

Then, one of them can be expressed as a linear combination of the other two.

Then, let

Rearranging them we get,

Here, let

x = l

y = m

–1 = n

We have,

Thus, if , and are coplanars, there exists scalar l, m and n (not all zero simultaneously zero) such that .

∴ necessary condition is proved.

(b). Sufficient condition: Let , and be three vectors such that there exists scalars l, m and n not all simultaneously zero such that .

Now, divide by n on both sides, we get

Here, we can see that

is the linear combination of and .

⇒ Clearly, , and are coplanar.

∴ sufficient condition is also proved.

Hence, proved.