Question

To show that: $\hat{i}-\hat{j}-6\hat{k};\hat{i}-3\hat{j}+4\hat{k};2\hat{i}-5\hat{j}+3\hat{k}$ are coplanar.

Answer 1

Let the given vector,

$\begin{array}{c}\overrightarrow{a}=\hat{i}-\hat{j}-6\hat{k}\\ \overrightarrow{b}=\hat{i}-3\hat{j}+4\hat{k}\\ \overrightarrow{c}=2\hat{i}-5\hat{j}+3\hat{k}\end{array}$

Vector $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ are coplanar

if $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=0$

Now,

$\begin{array}{c}\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]=\left|\begin{array}{ccc}1& -1& -6\\ 1& -3& 4\\ 2& -5& 3\end{array}\right|\\ \\ =1\left(-9+20\right)+1\left(3-8\right)-6\left(-5+6\right)\\ \\ =11-5-6=11-11\\ \\ =0\end{array}$

Hence, the given vectors are coplanar.