The sum of all values of , for which the points whose position vectors are i ^ - 2 j ^ + 3 k ^ , 2 i ^ - 3 j ^ + 4 k ^ , ( + 1 ) i ^ + 2 k ^ and 9 i ^ + ( - 8 ) j ^ + 6 k ^ are coplanar, is equal to [2023]

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Solution

Q.
The sum of all values of $α,$ for which the points whose position vectors are $\hat{i} - 2 \hat{j} + 3 \hat{k},$ $2 \hat{i} - 3 \hat{j} + 4 \hat{k},$ $(α + 1) \hat{i} + 2 \hat{k}$ and $9 \hat{i} + (α - 8) \hat{j} + 6 \hat{k}$ are coplanar, is equal to [2023]

1 6

2 - 2

3 2

4 4

Ans.
(3)

Let

$\vec{O A} = \hat{i} - 2 \hat{j} + 3 \hat{k},$ $\vec{O B} = 2 \hat{i} - 3 \hat{j} + 4 \hat{k},$ $\vec{O C} = (α + 1) \hat{i} + 2 \hat{k},$

$\vec{O D} = 9 \hat{i} + (α - 8) \hat{j} + 6 \hat{k}$

$∴$ $\vec{A B} = \hat{i} - \hat{j} + \hat{k},$ $\vec{A C} = α \hat{i} + 2 \hat{j} - \hat{k}$

$\vec{A D} = 8 \hat{i} + (α - 6) \hat{j} + 3 \hat{k}$

Now, $(\vec{A B} \times \vec{A C}) \cdot \vec{A D} = 0$ $\Rightarrow$ $| \begin{matrix} 1 & - 1 & 1 \\ α & 2 & - 1 \\ 8 & α - 6 & 3 \end{matrix} |$ $= 0$

$\Rightarrow 1 (6 + α - 6) + 1 (3 α + 8) + 1 (α^{2} - 6 α - 16) = 0$

$\Rightarrow α^{2} - 2 α - 8 = 0 \Rightarrow (α - 4) (α + 2) = 0$

$\Rightarrow$ $α = 4 or α = - 2$

$∴$ Required sum $= 4 - 2 = 2$

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