MHT CET · Physics · Mathematics in Physics

P and Q are two non-zero vectors inclined to each other at an angle $' θ^{'} .$ ‘p’ and ‘q’ are unit vectors along P and Q respectively. The component of Q in the direction of Q will be

A $P . Q$
B $\frac{P \times Q}{P}$
C $\frac{P . Q}{Q}$
D $p . q$

Verified Solution

Answer & Solution

Correct Answer

(A) $P . Q$

Step-by-step Solution

Detailed explanation

Let two vectors P and Q are represented by graph as below

Here,

Q_{p}

is a vector in the direction of P.
Then, from the right angle triangle, we get

c o s θ = \frac{Q_{p}}{Q}

….(i)

\Rightarrow Q_{p} = Q c o s θ

Also,

c o s θ = \frac{P . Q}{P Q} \Rightarrow \frac{Q_{p}}{Q} = \frac{P . Q}{P Q}

\Rightarrow Q_{p} = \frac{P . Q}{P}

….(ii)
As given that, P is the unit vector along P, then

\hat{P} = \frac{P}{P}

…(iii)
Putting the value of P from Eq.(iii) to Eq.(ii), we get

Q_{p} = P . Q

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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P and Q are two non-zero vectors inclined to each other at an angle 'θ'. ‘p’ and ‘q’ are unit vectors along P and Q respectively. The component of Q in the direction of Q will be

Step-by-step Solution

P and Q are two non-zero vectors inclined to each other at an angle $' θ^{'} .$ ‘p’ and ‘q’ are unit vectors along P and Q respectively. The component of Q in the direction of Q will be