Question

Prove that the following vectors are non-coplanar:
(i) $3\hat{i}+\hat{j}-\hat{k},2\hat{i}-\hat{j}+7\hat{k}\mathrm{and}7\hat{i}-\hat{j}+23\hat{k}$

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

Answer 1

(i) Let if possible the given vectors are coplanar. Then one of the given vector is expressible in terms of the other two.
We have,
$$\begin{gathered} 3\hat{i}+\hat{j}-\hat{k}=x(2\hat{i}-\hat{j}+7\hat{k})+y(7\hat{i}-\hat{j}+23\hat{k}). \\ =\hat{i}(2x+7y)+\hat{j}(-x-y)+\hat{k}(7x+23y). \\ \Rightarrow 2x+7y=3,x+y=-1,7x+23y=-1. \\ \mathrm{By}\mathrm{solving}\mathrm{the}\mathrm{first}\mathrm{two}\mathrm{equations},\mathrm{we}\mathrm{get} \\ \Rightarrow x=-2,y=1. \end{gathered}$$
Clearly these values of x and y does not satisfy the third equation.
Hence the given vectors are non-coplanar.


(ii) Let if possible the given vectors are coplanar. Then one of the given vector is expressible in terms of the other two.
We have,
$$\begin{gathered} \hat{i}+2\hat{j}+3\hat{k}=x(2\hat{i}+\hat{j}+3\hat{k})+y(\hat{i}+\hat{j}+\hat{k}). \\ =\hat{i}(2x+y)+\hat{j}(x+y)+\hat{k}(3x+y). \\ \Rightarrow 2x+y=1,x+y=2,3x+y=3. \\ \mathrm{By}\mathrm{solving}\mathrm{the}\mathrm{first}\mathrm{two}\mathrm{equation},\mathrm{we}\mathrm{get} \\ \Rightarrow x=-1,y=3. \end{gathered}$$
Clearly these values of x and y does not satisfy the third equation.
Hence the given vectors are non-coplanar.