Question

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 ______.

Options

• P · Q

• `("P" xx "Q")/"P"`

• `("P" * "Q")/"Q"`

• p · q

Answer 1

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 P · Q.

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

`cos theta = "Q"_"p"/"Q"`     ....(i)

`=> "Q"_"p" = "Q" cos theta`

Also, `cos theta = ("P" * "Q")/("PQ")`

`=> "Q"_"P"/"Q" = ("P" * "Q")/("PQ")`

`=> "Q"_"P" = ("P" * "Q")/"P"`     ....(ii)

As given that, `hat "P"` is the unit vector along with P, then

`hat "P" = "P"/"P"`     ....(iii)

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

`"Q"_"p" = hat"P" * "Q"`