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Question
For what value of 'a' the vectors \[2 \hat{i} - 3 \hat{j} + 4 \hat{k} \text{ and }a \hat{i} + 6 \hat{j} - 8 \hat{k}\] are collinear?
Solution
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, \[\overrightarrow{p} = \lambda \overrightarrow{q}\]
\[\Rightarrow 2 \hat{i} - 3 \hat{j} + 4 \hat{k} = \lambda \left( a \hat{i} + 6 \hat{j} - 8 \hat{k} \right)\]
\[ \Rightarrow 2 \hat{i} - 3 \hat{j} + 4 \hat{k} = a\lambda \hat{i} + 6\lambda \hat{j} - 8\lambda \hat{k} \]
\[\Rightarrow \lambda a = 2, 6\lambda = - 3\text{ and }- 8\lambda = 4\]
\[ \Rightarrow \lambda = - \frac{1}{2}\text{ and }a = - 4\]
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