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Applications of Cross Product

A vector c pe...

Question

A vector c perpendicular to the vectors $2 i + 3 j - k$ and $i - 2 j + 3 k$ satisfying $c \cdot (2 i - j + k) = - 6$ is

- 2 i + j - k

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2 i - j - \frac{4}{3} k

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- 3 i + 3 j + 3 k

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3 i - 3 j + 3 k

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Solution

The correct option is B $- 3 i + 3 j + 3 k$
$\begin{matrix} L e t C = C_{1} ˆ i + C_{2} ˆ j + C_{3} ˆ k A . T . Q (2 ˆ i + 3 ˆ j - ˆ k) . (C_{1} ˆ i + C_{2} ˆ j + C_{3} ˆ k) = 0 2 C_{1} + 3 C_{2} + C_{3} = 0............... (1) (ˆ i - 2 ˆ j + 3 ˆ k) . (C_{1} ˆ i + C_{2} ˆ j + C_{3} ˆ k) = 0 C_{1} - 2 C_{2} + 3 C_{3} = 0............... (2) (C_{1} ˆ i + C_{2} ˆ j + C_{3} ˆ k) . (2 ˆ i - ˆ j + ˆ k) = - 6 2 c_{1} - c_{2} + c_{3} = - 6............. (3) f r o m e q u a t i o n (1), (2), a n d (3) c_{1} = - 3, c_{2} = 3 c_{3} = 3 ∴ \to c = (- 3 ˆ i + 3 ˆ j + 3 ˆ k) \end{matrix}$

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