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Question
Show that the vectors \[2 \hat{i} - 3 \hat{j} + 4 \hat{k}\text{ and }- 4 \hat{i} + 6 \hat{j} - 8 \hat{k}\] are collinear.
Solution
Given the position vectors \[2 \hat{i} - 3 \hat{j} + 4 \hat{k}\] and \[- 4 \hat{i} + 6 \hat{j} - 8 \hat{k}\]
Let \[\vec{a} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k}\] and \[\vec{b} = - 4 \hat{i} + 6 \hat{j} - 8 \hat{k}\]
Then,
\[\vec{b} = - 4 \hat{i} + 6 \hat{j} - 8 \hat{k} \]
\[ = - 2\left( 2 \hat{i} - 3 \hat{j} + 4 \hat{k} \right)\]
\[ = - 2 \vec{a} \]
Hence, \[\vec{a} , \vec{b}\] are collinear.
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