Question

Check whether the vector $4\hat{i}+13\hat{j}-18\hat{k}$ is a linear combination of the vectors $\hat{i}-2\hat{j}+3\hat{k}$ and $2\hat{i}+3\hat{j}-4\hat{k}$

Answer 1

If a vector is a linear combination of other two vector it means all three vectors are coplanar.

$\therefore \overrightarrow{a}\cdot (\overrightarrow{b}\times \overrightarrow{c})=0$ (To check)

or $\left|\begin{array}{ccc}4& 13& -18\\ 1& -2& 3\\ 2& 3& -4\end{array}\right|$

$=4(8-9)-13(-4-6)-18(3+4)$

$=-4+130-126=0$

Hence vector are coplanar.