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Question
If the vectors \[4 \hat { i} + 11 \hat {j} + m \hat {k} , 7 \hat { i} + 2 \hat { j} + 6 \hat {k} \text { and } \hat {i} + 5 \hat {j} + 4 \hat {k}\] are coplanar, then m =
Options
0
38
-10
10
Solution
10
Let:
\[ \vec{a} = 4 \hat {i} + 11 \hat {j} + m \hat {k} \]
\[ \vec{b} = 7 \hat {i} + 2 \hat {j} + 6 \hat {k} \]
\[ \vec{c} = \hat {i} + 5 \hat {j} + 4 \hat {k} \]
\[\text { We know that vectors }\vec{a} , \vec{b} \text { and }\vec{c} \text { are coplanar iff their scalar triple product is zero, i . e }. \left[ \vec{a} \vec{b} \vec{c} \right] = 0\]
\[ \Rightarrow \begin{vmatrix}4 & 11 & m \\ 7 & 2 & 6 \\ 1 & 5 & 4\end{vmatrix} = 0 \]
\[ \Rightarrow 4\left( 8 - 30 \right) - 11\left( 28 - 6 \right) + m\left( 35 - 2 \right) = 0\]
\[ \Rightarrow - 88 - 242 + 33m = 0\]
\[ \Rightarrow 33m = 330 \]
\[ \therefore m = 10\]
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