Question

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

Answer 1

Hint : Coplanar vectors are the vectors which lie on the same plane. To prove necessary conditions, assume that the vectors are coplanar and using that, prove $l\overrightarrow a + m\overrightarrow b + n\overrightarrow c = \overrightarrow 0$ . To prove sufficient condition, assume $l\overrightarrow a + m\overrightarrow b + n\overrightarrow c = \overrightarrow 0$ and using that, prove the vectors are coplanar.

Complete step-by-step answer :
Necessary condition:
Let $\overrightarrow a ,\overrightarrow b ,\overrightarrow c$ be three coplanar vectors.
Coplanar vectors mean the vectors that lie on the same plane.
Now, since they are coplanar, we can write them as a linear combination of one of the vectors. We can explain this by following two conditions.
1. Let all three vectors be parallel to each other. Then by the property of parallel vectors,
$\overrightarrow a = k\overrightarrow b = \lambda \overrightarrow c$
Where $k$ and $\lambda$ are some constants.
Hence, we can write
$\overrightarrow a - k\overrightarrow b = \overrightarrow 0$ and $\overrightarrow a - \lambda \overrightarrow c = \overrightarrow 0$
By adding both of them, we get
$2\overrightarrow a - k\overrightarrow b - \lambda \overrightarrow c = \overrightarrow 0$
By substituting $l = 2,m = - k,n = - \lambda$ , we get
$\Rightarrow l\overrightarrow a + m\overrightarrow b + n\overrightarrow c = \overrightarrow 0$
2. If they are not parallel to each other. Then they will intersect each other. So by using triangular law of addition, we can say that one of the vectors is written as the sum of the other two vectors.
Thus say,
$\overrightarrow a = x\overrightarrow b + y\overrightarrow c$
Where, $x$ and $y$ are some constants.
Rearranging it we can write
$\overrightarrow a - x\overrightarrow b - y\overrightarrow c = \overrightarrow 0$
By substituting $l = 1,m = - x,n = - y$ , we get
$\Rightarrow l\overrightarrow a + m\overrightarrow b + n\overrightarrow c = \overrightarrow 0$
Hence we can conclude that $l\overrightarrow a + m\overrightarrow b + n\overrightarrow c = \overrightarrow 0$ , is the necessary condition to prove vectors $\overrightarrow a ,\overrightarrow b ,\overrightarrow c$ are coplanar.
Sufficient condition:
If vectors $\overrightarrow a ,\overrightarrow b ,\overrightarrow c$ can be written as $l\overrightarrow a + m\overrightarrow b + n\overrightarrow c = \overrightarrow 0$
Then by rearranging it we can write
$\Rightarrow l\overrightarrow a = - m\overrightarrow b - n\overrightarrow c$
By dividing both the side by $l$ , we get
$\Rightarrow \overrightarrow a = - \dfrac{m}{l}\overrightarrow b - \dfrac{n}{l}\overrightarrow c$
Which further can be written as
$\overrightarrow a = p\overrightarrow b + q\overrightarrow c$
Where,
$p = - \dfrac{m}{l}$ and $q = - \dfrac{n}{l}$ are two constants.
Thus, we can write $\overrightarrow a$ as a linear combination of $\overrightarrow b$ and $\overrightarrow c$ . Thus by the property of coplanar vectors, vectors, $\overrightarrow a ,\overrightarrow b ,\overrightarrow c$ are coplanar.
Thus,
$\Rightarrow l\overrightarrow a + m\overrightarrow b + n\overrightarrow c = \overrightarrow 0$ is the sufficient condition to prove that vectors, $\overrightarrow a ,\overrightarrow b ,\overrightarrow c$ are coplanar.

Note : You need to know the difference between necessary and sufficient condition. Necessary condition is the condition without which you cannot prove the theorem or statement that you are arguing to be true. And sufficient condition is the condition which is enough and does not need any other supporting condition to prove the theorem or statement that you are arguing to be true.