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Q.

Let the position vectors of the points P and Q be $4 \hat{i} + \hat{j} + λ \hat{k} and 2 \hat{i} - \hat{j} + λ \hat{k},$ respectively. Vector $\hat{i} - \hat{j} + 6 \hat{k}$ is perpendicular to the plane containing the origin and the points P and Q. Then $λ$ equals

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a

1

b

-1/2

c

none of these

d

1/2

answer is A.

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Detailed Solution

A vector perpendicular to the plane of O, P and Q is $\overset{⟶}{O P} \times \overset{⟶}{O Q}$ .

Now, $\overset{⟶}{O P} \times \overset{⟶}{O Q} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & 1 & λ \\ 2 & - 1 & λ \end{matrix}|$

$= 2 λ \hat{i} - 2 λ \hat{j} - 6 \hat{k}$

Therefore, $\hat{i} - \hat{j} + 6 \hat{k}$ is parallel to $2 λ \hat{i} - 2 λ \hat{j} - 6 \hat{k}$

Hence,

$\begin{array}{l} \frac{1}{2 λ} = \frac{- 1}{- 2 λ} = \frac{6}{- 6} \\ λ = - \frac{1}{2} \end{array}$

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