Two objects `A and B` are moving along the directions as shown in (Fig. 5.77). Find the magnitude and direction of the relative velocity of `B w.r.t. A`.
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Explanation

Analytically :
`vec v_A = 10 hat i m//s^-1, vecv_B = 20 cos 30^@ hat i + 20 sin 30^@ hat j`
=`(10 sqrt(3) hat i + 10 hat j)) m//s^-1`
Relative velocity of `B` w.r.t. `A`
`vec v_(B//A) = vec v_B - vec v_A = 10 (sqrt(3 - 1)^2 + 1^2)) = 10 sqrt(5 - 2 sqrt(3)) m s^-1`
`tan prop = (10)/(10(sqrt(3 - 1))) = (1)/(sqrt(3)- 1)`
Graphically : Velocity of `B` with respect to `A`
`vec v_(B//A) = vec v_B + (- vec v_A)`
From (Fig. 5.78), `NS = MP = 20 sin 30^@ = 10`
and `ON = OM - NM = 20 cos 30^@ - 10 = 10 (sqrt(3 - 1))`
`|vec v_(B//A)| = sqrt(ON^2 + NS^2)) = 10 sqrt((3 - 1)^2 + (1)^2)`
=`10 sqrt(5 - 2 sqrt(3)) ms^-1`
`tan prop = (NS)/(ON) = (10)/(10 (sqrt(3 - 1))) = (1)/(sqrt(3) - 1))`.