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Question
Show of the following triad of vector is coplanar:
\[\vec{a} = \hat {i} + 2 \hat{j} - \hat {k} , \vec{b} = 3 \hat {i} + 2 \hat{j} + 7 \hat {k} , \vec{c} = 5 \hat {i} + 6 \hat { j} + 5 \hat {k}\]
Solution
Given:
\[ \vec{a} = \hat{i} + 2 \hat {j} - \hat {k} \]
\[ \vec{b} = 3\hat{ i} + 2 \hat {j} + 7 \hat {k} \]
\[ \vec{c} = 5 \hat {i} + 6 \hat {j} + 5 \hat {k} \]
\[\text{We know that three vectors} \vec{a} , \vec{b} , \vec{c} \text{are coplanar iff their scalar triple product is zero, i . e }. \left[ \vec{a} \vec{b} \vec{c} \right] = 0\]
Here,
\[\left[ \vec{a} \vec{b} \vec{c} \right] = \begin{vmatrix}1 & 2 & - 1 \\ 3 & 2 & 7 \\ 5 & 6 & 5\end{vmatrix} = 1 \left( 10 - 42 \right) - 2\left( 15 - 35 \right) - 1\left( 18 - 10 \right) = 0\]
Hence, the given vectors are coplanar .
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