Question

find the magnitude and the director of resultant of two vectors a and b in terms of their magnitude and the angle tetha between them​

Answer 1

Explanation:

Let OP and OQ represent the two vectors A and B making an angle θ. Then, using the parallelogram method of vector addition, OS represents the resultant vector R.

R= A+B

SN is normal to OP and OP and PM is normal to OS.

From the geometry of the figure.

(OS)2=(ON)2+(SN)2

but ON = OP + PN = A + Bcosθ

SN = Bsinθ

(OS)2=(A+Bcosθ)2+(Bsinθ)2

or, R2=A2+B2+2ABcosθ−−−−−−−−−−−−−−−−−√.............(4.24a)

In Δ OSN, SN= OSsinα=Rsinα, and in ΔPSN, SN= PS sinθ

Therefore, Rsinα=Bsinθ

or, Rsinθ=Bsinα................(4.24b)

Similarly,

PM= Asinα=Bsinβ

or Asinβ=Bsinα ............(4.24c)

Combining Eqs. we get,

Rsinθ=Asinβ=Bsinα................(4.24d)

Using Eqs, we get

sinalph=BRsinθ ...............(4.24 e)

Where R is given by Eq.

or tanα=SNOP+PN=BsinθA+Bcosθ...............(4.24f)

Eqs. (4.24a) givestjhe magnitude of the resultant and Eqs. (4.24e) and (4.24f) its direction. Equation (4.24a) is known as the law of cosines and Eq. (4.24d) as the law of sines.