Prove that a necessary and sufficient condition for three vectors vector a, vector b and vector c to be coplanar is that there exist scalars

Question

Prove that a necessary and sufficient condition for three vectors \(\vec a,\vec b\) and \(\vec c\) to be coplanar is that there exist scalars l, m, n not zero simultaneously such that\(l\vec a+m\vec b+n\vec c=\vec 0\).

Answer 1

Given: The vectors  \(\vec a,\,\vec b\) and \(\vec c\)

To Prove: (a) Necessary condition: The vectors  \(\vec a,\,\vec b\) and \(\vec c\) and will be coplanar if there exist scalar l, m, n not all zero simultaneously such that \(l\vec a+m\vec b+n\vec c=0.\)

(b). Sufficient condition: Let  \(\vec a,\,\vec b\) and \(\vec c\) there exist scalar l, m, n not all zero simultaneously such that \(l\vec a+m\vec b+n\vec c=0.\)

Proof:

(a). Necessary condition: Let  \(\vec a,\,\vec b\) and \(\vec c\) are three coplanar vectors.

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

Then, let \(\vec c=\text xa+y\vec b\)

Rearranging them we get,

\(\text xa+y\vec b-\vec c=0\)

Here, let

x = l

y = m

–1 = n

We have,

\(l\vec a+m\vec b+n\vec c=0\)

Thus, if  \(\vec a,\,\vec b\) and \(\vec c\) are coplanars, there exists scalar l, m and n (not all zero simultaneously zero) such that \(l\vec a+m\vec b+n\vec c=0\)

∴ necessary condition is proved.

(b). Sufficient condition: Let  \(\vec a,\,\vec b\) and \(\vec c\) be three vectors such that there exists scalars l, m and n not all simultaneously zero such that \(l\vec a+m\vec b+n\vec c=0\)

Now, divide by n on both sides, we get

Here, we can see that

\(\vec c\) is the linear combination \(\vec a\) and \(\vec b\)

⇒ Clearly, \(\vec a,\,\vec b\) and \(\vec c\) are coplanar.

∴ sufficient condition is also proved.

Hence, proved.