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Question
Show of the following triad of vector is coplanar:
\[\hat{a} = \hat{i} - 2 \hat {j} + 3 \hat {k} , \hat {b} = - 2 \hat {i} + 3 \hat {j} - 4 \hat { k}, \hat {c} = \hat { i} - 3 \hat { j} + 5 \hat { k }\]
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Solution
Given:
\[ \vec{a} = \hat { i} - 2 \hat{j} + 3 \hat{k} \]
\[ \vec{b} = - 2 \hat{i} + 3 \hat{j} - 4 \hat{k} \]
\[ \vec{c} = \hat{i} - 3 \hat{j} + 5\hat{ k} \]
\[\text { We know that three vectors }\vec{a} , \vec{b} , \vec{c} \text { are coplanar iff their scalar triple product is zero, i . e .} \left[ \vec{a} \vec{b} \vec{c} \right] = 0 . \]
Here,
\[\left[ \vec{a} \vec{b} \vec{c} \right] = \begin{vmatrix}1 & - 2 & 3 \\ - 2 & 3 & - 4 \\ 1 & - 3 & 5\end{vmatrix} = 1\left( 15 - 12 \right) + 2\left( - 10 + 4 \right) + 3\left( 6 - 3 \right) = 0\]
Hence, the given vectors are coplanar .
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