Test: Vector Addition - Analytical Method (NCERT) - NEET MCQ

A vector, its magnitude and the angle between two vectors do not depend on the choice of the orientation of the coordinate axes. So angle between are independent of the orientation of the coordinate axes. But the quantity A_x + B_y depends upon the magnitude of the components along x and y axes, so it will change with change in coordinate axes.

Let OP and OQ represent two vectorsmaking an angle (α + β). Using the parallelogram method of vector addition,
Resultant vector, 
SN is normal to OP and PM is normal to OS.
From the geometry of the figure,
OS² = ON² + SN² = (OP + PN)² + SN² = (A + Bcos(α + β))² + (Bsin(α + β))²

R² = A² + B² + 2ABcos(α + β)

In ΔOSN, SN = OSsinα = Rsinα and in ΔPSN,SN = PSsin(α + β) = Bsin(α + β)
Rsinα = Bsin(α + β) or R/sin(α + β)= B/sinα
Similarly,
PM = Asinα = Bsinβ
A/sinβ = B/sinα
Combining (i) and (ii), we get
R/sin(α + β) = A/sinβ = B/sinα
From eqn, (iii), Rsinβ = Asin(α + β)

Let θ be the angle between the vectors

velocity of the water current v_c​ = 10 km/s

velocity of the motorboat v_b ​= 25 km/h

angle between north and south east is 120^o

The resultant velocity of the boat is 22km/h⁻¹