Question

Prove that the following vectors are coplanar:
(i) $2\hat{i}-\hat{j}+\hat{k},\hat{i}-3\hat{j}-5\hat{k}\mathrm{and}3\hat{i}-4\hat{j}-4\hat{k}$

(ii) $\hat{i}+\hat{j}+\hat{k},2\hat{i}+3\hat{j}-\hat{k}\mathrm{and}-\hat{i}-2\hat{j}+2\hat{k}$

Answer 1

(i) Given the vectors $P\left(2\hat{i}-\hat{j}+\hat{k}\right),Q\left(\hat{i}-3\hat{j}-5\hat{k}\right)$ and $R\left(3\hat{i}-4\hat{j}-4\hat{k}\right)$.
We know the three vectors are coplanar if one of them is expressible as a linear combination of the other two. Let,
$$\begin{gathered} 2\hat{i}-\hat{j}+\hat{k}=x\left(\hat{i}-3\hat{j}-5\hat{k}\right)+y\left(3\hat{i}-4\hat{j}-4\hat{k}\right). \\ =\hat{i}\left(x+3y\right)+\hat{j}\left(-3x-4y\right)+\hat{k}\left(-5x-4y\right). \end{gathered}$$
$\Rightarrow x+3y=2,-3x-4y=-1,-5x-4y=1$ [Equating the coefficients of $\hat{i},\hat{j},\hat{k}$ respectively]
Solving first two of these equation, we get $x=-1,y=1$. Clearly these two values satisfy the third equation.
Hence, the given vectors are coplanar.

(ii) Given the vectors $P\left(\hat{i}+\hat{j}+\hat{k}\right),Q\left(2\hat{i}+3\hat{j}-\hat{k}\right)$ and $R\left(-\hat{i}-2\hat{j}+2\hat{k}\right)$.
We know the three vectors are coplanar if one of them is expressible as a linear combination of the other two. Let,
$$\begin{gathered} \hat{i}+\hat{j}+\hat{k}=x\left(2\hat{i}+3\hat{j}-\hat{k}\right)+y\left(-\hat{i}-2\hat{j}+2\hat{k}\right). \\ =\hat{i}\left(2x-y\right)+\hat{j}\left(3x-2y\right)+\hat{k}\left(-x+2y\right). \end{gathered}$$
$\Rightarrow 2x-y=1,3x-2y=1,-x+2y=1$ [ Equating the coefficients of $\hat{i},\hat{j},\hat{k}$ respectvely]
Solving first two of these equation , we get $x=1,y=1$. Clearly these two values satisfy the third equation.
Hence, the given vectors are coplanar.