Let O be the origin and the position vector of the point P be ? - i ^ - 2 j ^ + 3 k ^ . If the position vectors of the points A, B and C are - 2 i ^ + j ^ - 3 k ^ , 2 i ^ + 4 j ^ - 2 k ^ and - 4 i ^ + 2 j ^ - k ^ respectively, then the projection o

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Solution

Q.
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 $\vec{O P}$ on a vector perpendicular to the vectors $\vec{A B}$ and $\vec{A C}$ is [2023]

1 $\frac{8}{3}$

2 $\frac{7}{3}$

3 $3$

4 $\frac{10}{3}$

Ans.
(3)

$We know that$

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

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

$Now, \vec{A B} \times \vec{A C} =$ $| \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & 3 & 1 \\ - 2 & 1 & 2 \end{matrix} |$ $= 5 \hat{i} - 10 \hat{j} + 10 \hat{k}$

$Also, \vec{O P} = - \hat{i} - 2 \hat{j} + 3 \hat{k}$

$Projection of \vec{O P} on the vector perpendicular to the vectors \vec{A B} and \vec{A C} is given by$

$\frac{\vec{O P} \cdot (\vec{A B} \times \vec{A C})}{| \vec{A B} \times \vec{A C} |} = \frac{(- \hat{i} - 2 \hat{j} + 3 \hat{k}) (5 \hat{i} - 10 \hat{j} + 10 \hat{k})}{\sqrt{5^{2} + {(- 10)}^{2} + {(10)}^{2}}}$

$= \frac{- 5 + 20 + 30}{\sqrt{225}} = \frac{45}{15} = 3$

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