Let O be the origin and the position vector of the point P be −i^−2j^+3k^. If the position vectors of the points A, B and C are −2i^+j^−3k^,2i^+4j^−2k^ and −4i^+2j^−k^, respectively, then the projection of the vector OP¯ on a vector perpendicular to the vectors AB¯ and AC¯ is

AB→=OB→−OA→=4i^+3j^+k^ AC→=OC→−OA→=−2i^+j^+2k^AB→×AC→=5i^−10j^+10k^OP→=−i^−2j^+3k^ The projection of the vector OP¯ on a vector perpendicular to the vectors AB¯ and AC¯ isOP¯⋅AB¯×AC¯|AB¯×AC¯|=3

Let O be the origin and the position vector of the point P be −i^−2j^+3k^. If the position vectors of the points A, B and C are −2i^+j^−3k^,2i^+4j^−2k^ and −4i^+2j^−k^, respectively, then the projection of the vector OP¯ on a vector perpendicular to the vectors AB¯ and AC¯ is

AB→=OB→−OA→=4i^+3j^+k^ AC→=OC→−OA→=−2i^+j^+2k^AB→×AC→=5i^−10j^+10k^OP→=−i^−2j^+3k^ The projection of the vector OP¯ on a vector perpendicular to the vectors AB¯ and AC¯ isOP¯⋅AB¯×AC¯|AB¯×AC¯|=3

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Let O be the origin and the position vector of the point P be $- \hat{i} - 2 \hat{j} + 3 \hat{k}$ . If the position vectors of the points A, B and C are $- 2 \hat{i} + \hat{j} - 3 \hat{k}, 2 \hat{i} + 4 \hat{j} - 2 \hat{k} and - 4 \hat{i} + 2 \hat{j} - \hat{k}$ , respectively, then the projection of the vector $\bar{OP}$ on a vector perpendicular to the vectors $\bar{AB} and \bar{AC}$ is

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$\frac{8}{3}$

$\frac{10}{3}$

$\frac{7}{3}$

answer is B.

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Detailed Solution

$\vec{A B} = \vec{O B} - \vec{O A} = 4 \hat{i} + 3 \hat{j} + \hat{k} \begin{array}{l} \vec{A C} = \vec{O C} - \vec{O A} = - 2 \hat{i} + \hat{j} + 2 \hat{k} \\ \vec{A B} \times \vec{A C} = 5 \hat{i} - 10 \hat{j} + 10 \hat{k} \\ \vec{O P} = - \hat{i} - 2 \hat{j} + 3 \hat{k} \end{array}$