JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Vector Algebra

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

O P

on a vector perpendicular to the vectors

A B

and

A C

Answer: (A)

Solution

First, find the vectors

A B

and

A C

A B = O B - 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}

A C = O C - 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}

The vector perpendicular to both

A B

and

A C

is their cross product:

A B \times A C = 5 \hat{i} - 10 \hat{j} + 10 \hat{k}

The projection of

O P

on this perpendicular vector is:

Projection = \frac{O P \cdot ( A B \times A C )}{∣ A B \times A C ∣} = \frac{( - 1 ) ( 5 ) + ( - 2 ) ( - 10 ) + ( 3 ) ( 10 )}{25 + 100 + 100} = \frac{- 5 + 20 + 30}{15} = \frac{45}{15} = 3

JEE Main · 2023 · 10 Apr 2023 · Shift 1 · Single correct option