Question

If four points A, B, C and D with position vectors 4 \[\hat { i} +3\] \[\hat { j} +3\] \[\hat { k} ,5\] \[\hat { i} +\] \[x\hat { j} +7\] \[\hat { k} ,5\] \[\hat { i} +3\] \[\hat { j}\] and \[7 \hat{i} + 6 \hat{j} + \hat{k}\] respectively are coplanar, then find the value of x.

Answer 1

Let  \[\overrightarrow{OA} = 4 \hat { i } + 3 \hat { j } + 3 \hat { k} , \overrightarrow{OB} = 5 \hat { i} + x\hat{ j} + 7 \hat{k} , \overrightarrow{OC} = 5 \hat { i} + 3 \hat { j }\] and  \[\overrightarrow{OD} = 7\hat {  i} + 6 \hat { j} + \hat { k }\].

\[\therefore \overrightarrow{AB} = \left( 5 \hat { i } + x \hat { j} + 7 \hat { k} \right) - \left( 4 \hat { i} + 3 \hat { j} + 3 \hat { k} \right) =\hat { i} + \left( x - 3 \right) \hat{j} + 4 \hat { k }\]

\[\overrightarrow{AC} = \left( 5 \hat { i} + 3 \hat { j} \right) - \left( 4 \hat { i} + 3 \hat { j} + 3 \hat { k} \right) = \hat {i} - 3 \hat {k}\]

\[\overrightarrow{AD} = \left( 7 \hat { i} + 6 \hat{j } + \hat { k} \right) - \left( 4 \hat {i} + 3 \hat { j} + 3 \hat {k} \right) = 3 \hat { i} + 3 \hat {j} - 2 \hat {k}\]

Since the given four points are coplanar, so the vectors  \[\overrightarrow{AB} , \overrightarrow{AC}\] and  \[\overrightarrow {AD}\] are also coplanar.

\[\therefore \begin{bmatrix}\overrightarrow{AB} & \overrightarrow{AC} & \overrightarrow{AD}\end{bmatrix} = 0\]

\[\Rightarrow \begin{vmatrix}1 & x - 3 & 4 \\ 1 & 0 & - 3 \\ 3 & 3 & - 2\end{vmatrix} = 0\]

\[ \Rightarrow 1\left( 0 + 9 \right) - \left( x - 3 \right)\left( - 2 + 9 \right) + 4\left( 3 - 0 \right) = 0\]

\[ \Rightarrow 9 - 7x + 21 + 12 = 0\]

\[ \Rightarrow 7x = 42\]

\[ \Rightarrow x = 6\]

Thus, the value of x is 6.