Find the value of , if four points with position vectors 3 i + 6 … - Vidyadip

Question

Find the value of $\lambda,$ if four points with position vectors $3\hat{\text{i}} + 6\hat{\text{j}} + 9\hat{\text{k}}, \hat{\text{i}} + 2\hat{\text{j}} + 3\hat{\text{k}}, 2\hat{\text{i}} + 3\hat{\text{j}} + \hat{\text{k}} \text{ and } 4\hat{\text{i}} + 6\hat{\text{j}} + \lambda\hat{\text{k}}$are coplanar.

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Answer

Given points, A, B, C, D are coplanar, if the
vectors $\vec{\text{AB}} , \vec{\text{AC}} \text{ and } \vec{\text{AD}}$ are coplanar, i.e.
$\vec{\text{AB}} = -2\hat{\text{i}} - 4\hat{\text{j}} - 6\hat{\text{k}}, \vec{\text{AC}} = - \hat{\text{i}} - 3\hat{\text{j}} - 8\hat{\text{k}}, \vec{\text{AD}} = \hat{\text{i}} + (\lambda - 9) \hat{\text{k}}$
are coplanar
$\text{i.e.,}\begin{vmatrix} -2 & -4 & -6 \\ -1 & -3 & -8 \\ 1 & 0 & \lambda - 9 \end{vmatrix} = 0$
$- 2[-3\lambda +27] + 4 [ -\lambda + 17] -6 (3) = 0$
$\Rightarrow \lambda = 2.$

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