Clock time

We should know that there is a formula for dot product of any two vectors according to which the dot product of two vectors $P^{\rightarrow},Q^{\rightarrow}$ with angle $\theta$ between them is given by $P^{\rightarrow}.Q^{\rightarrow}=PQ\cos \theta$ where $P,Q$ are the magnitudes of the vectors.

A clock is a circle that has a complete whole angle of $360^{\circ}$ from the centre of the clock. Now, if the minute hand is fixed at the $12$ position, then the hour hand makes an additional $30^{\circ}$ per move through an hour position of hour hand. Let the two vectors (minute hand and hour hand) be $A^{\rightarrow}$ and $B^{\rightarrow}$ respectively.

$360^{\circ}........\quad 12 \text{ hrs }$

$2 hrs ........ \quad \frac{360}{12}\times 2 = \boxed{60^{\circ}}$

So, the angle between the $2$ vectors is $\theta = 60^{\circ}$ . Now, the magnitudes are $A=4$ and $B=2$ .

So, $A^{\rightarrow}.B^{\rightarrow}=AB\cos \theta = 4\times 2\times \cos 60^{\circ}=8\times \frac{1}{2}=\boxed{4}$

DOT PRODUCT OF TWO VECTORS IS a · b = |a| × |b| × cos(θ) where, |a| means the magnitude (length) of vector a

Here theta is 60 degrees, hence a.b = 4x2 Cos 60 degrees = 4x2X0.5 = 4