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Question
Find the value of λ for which the four points with position vectors
\[-\hat { j} - \hat {k} , 4 \hat {i} + 5 \hat {j} + \lambda \hat {k} , 3 \hat {i} + 9 \hat {j} + 4 \hat {k} \text { and } - 4 \hat {i} + 4 \hat {j} + 4 \hat{k}\]
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Solution
Let A, B, C and D be the given points . Then,
\[ \overrightarrow{AB} = (4 \hat{i} + 5 \hat {j} + \lambda \hat {k} ) - ( 0 \hat {i} - \stackrel\frown {j}- \stackrel\frown {k} ) = 4 \hat {i} + 6 \hat {j}+ (\lambda + 1) \hat {k} \]
\[ \overrightarrow{AC} = (3 \hat {i} + 9 \hat {j}+ 4 \hat {k}) - ( 0 \hat {i} - \hat {j} - \hat {k} ) = 3 \hat {i} + 10 \hat {j} + 5 \hat {k} \]
\[ \overrightarrow{AD} = ( - 4 \hat {i} + 4 \hat {j} + 4 \hat {k} ) - (0 \hat {i} - \hat {j} - \hat {k} ) = - 4 \hat {i} + 5 \hat {j} + 5 \hat {k} \]
\[\text{The given points are coplanar iff vectors }\overrightarrow{AB} , \overrightarrow{AC} , \overrightarrow{AD}\text { are coplanar} . \]
\[\text {Now,} \overrightarrow{AB} ,\overrightarrow{AC} , \overrightarrow{AD } \text { are coplanar} . \]
\[ \Rightarrow \begin{bmatrix}\overrightarrow{AB} & \overrightarrow{AC} & \overrightarrow{AD}\end{bmatrix} = 0\]
\[ \Rightarrow \begin{vmatrix}4 & 6 & (\lambda + 1) \\ 3 & 10 & 5 \\ - 4 & 5 & 5\end{vmatrix} = 0\]
\[ \Rightarrow 4(50 - 25) - 6 (15 + 20) + (\lambda + 1)(15 + 40) = 0\]
\[ \Rightarrow 100 - 210 + 55\lambda + 55 = 0\]
\[ \Rightarrow 55\lambda = 55\]
\[ \Rightarrow \lambda = 1\]
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