How we do addition of two vectors? Give some example.

Let and represent the two vectors and , making an angle θ.

For right-angled triangle ONS,

OS²= ON²+ SN²

However,ON = OP + PN = A+ B cosθ

SN = B sinθ

OS²= (A + Bcosθ)²+ (Bsinθ)²

⇒ R²= A²+ B² + 2ABcosθ … (i)

In ΔOSN, SN = OS sin α= R sin α

In ΔPSN, SN = PS sinθ= B sinθ

⇒

Similarly,

PM = A sinα= B sinβ

Combining equations (ii) and (iii), we obtain

⇒

Using equation (iv), we obtain

Where ‘R’ is given by equation (i)

⇒ 

Equation (i) gives the magnitude of the resultant and equation (v) and (vi) its directions.

Equation (i) is known as the law of cosines and equation (iv) as the law of sines.


 

Example − Two forces 10 N and 15 N are acting at an angle of 120°between them. Find the resultant force in magnitude and direction.

Solution

Here, A = 10 N, B = 15 N

θ = 120°; R =?; α= ?

⇒ 

⇒ 

⇒ 

⇒ R= 13.2 N

⇒ 

⇒ 

⇒ 

⇒ 

⇒ α = 76°