If the four points, whose position vectors are 3 i ^ - 4 j ^ + 2 k ^ , i ^ + 2 j ^ - k ^ , - 2 i ^ - j ^ + 3 k ^ and 5 i ^ - 2 j ^ + 4 k ^ are coplanar, then is equal to [2023]

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Solution

Q.
If the four points, whose position vectors are $3 \hat{i} - 4 \hat{j} + 2 \hat{k},$ $\hat{i} + 2 \hat{j} - \hat{k},$ $- 2 \hat{i} - \hat{j} + 3 \hat{k}$ and $5 \hat{i} - 2 α \hat{j} + 4 \hat{k}$ are coplanar, then $α$ is equal to [2023]

1 $\frac{107}{17}$

2 $\frac{73}{17}$

3 $- \frac{73}{17}$

4 $- \frac{107}{17}$

Ans.
(2)

$Four points with position vectors \vec{a}, \vec{b}, \vec{c}, \vec{d} respectively are coplanar iff [\vec{b} - \vec{a}, \vec{c} - \vec{a}, \vec{d} - \vec{a}] = 0$

$\vec{b} - \vec{a} = - 2 \hat{i} + 6 \hat{j} - 3 \hat{k}, \vec{c} - \vec{a} = - 5 \hat{i} + 3 \hat{j} + \hat{k}$

$\vec{d} - \vec{a} = 2 \hat{i} + (- 2 α + 4) \hat{j} + 2 \hat{k}$

$| \begin{matrix} - 2 & 6 & - 3 \\ - 5 & 3 & 1 \\ 2 & - 2 α + 4 & 2 \end{matrix} |$ $= 0$

$\Rightarrow - 4 - 4 α + 72 - 30 α + 78 = 0$

$\Rightarrow - 34 α + 146 = 0$

$\Rightarrow$ $α = \frac{73}{17}$

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