Question

For what value of 'a' the vectors $2\hat{i}-3\hat{j}+4\hat{k}\mathrm{and}a\hat{i}+6\hat{j}-8\hat{k}$ are collinear?

Answer 1

Given: Two vectors , let $\overrightarrow{p}=$ $2\hat{i}-3\hat{j}+4\hat{k}$ and $\overrightarrow{q}=$ $a\hat{i}+6\hat{j}-8\hat{k}$
Since the given vectors are collinear, we have,
$\overrightarrow{p}=\lambda \overrightarrow{q}$
$$\begin{gathered} \Rightarrow 2\hat{i}-3\hat{j}+4\hat{k}=\lambda \left(a\hat{i}+6\hat{j}-8\hat{k}\right) \\ \Rightarrow 2\hat{i}-3\hat{j}+4\hat{k}=a\lambda \hat{i}+6\lambda \hat{j}-8\lambda \hat{k} \end{gathered}$$

$$\begin{gathered} \Rightarrow \lambda a=2,6\lambda =-3\mathrm{and}-8\lambda =4 \\ \Rightarrow \lambda =-\frac{1}{2}\mathrm{and}a=-4 \end{gathered}$$