Question

A vector of length l is turned through the angel x about its tail. What is the change in the position vector of its head?

Answer 1

Answer:

2l sin²( x / 2 ).

Explanation:

Let the initial position of that vector be $\vec{r_1}$ and final position be $\vec{r_2}$.

Here,

Angle between the vectors is x.

Also,

| $\vec{r_1}$ | = | $\vec{r_2}$ | = l

Change in position ( using vector laws ) :

= > √[ l² + l² - 2l.l.cosx ]

= > √( 2l² - 2l²cosx )

= > l√[ 2( 1 - cosx )

= > l√[ 2{ 2 sin² ( x / 2 ) } ] { 1 - cosx = 2sin²( x / 2 ) }

= > l√( 4( sin² ( x / 2 )))

= > 2l sin²( x / 2 )

Hence, change in position vector is 2l sin²( x / 2 ).