Question

Two vectors A and B are inclined to each other at an angle Φ . Derive an expression for magnitude and direction of their resultant .​

Answer 1

Explanation:-

Two vectors A and B are inclined to each other at an angle $\theta$.

Suppose the magnitude of $\vec{A }= a$ and magnitude of $\vec{B} = b$.

We have to find :-

$| \vec{A}+\vec{B}|$

Derivation :-

In $\triangle{DEB}$

$Cos \theta = \dfrac{BE}{DB}$

→$BE = DB Cos \theta ; DB = b$

→$BE = b Cos \theta$

Similarly , $DE = b Sin \theta$

In $\triangle {ADE} , \angle{E} = 90^{\circ}$

→$AD^2 = AE^2 + DE^2$

→$AD^2 = ( AB + BE)^2 + DE^2$

→$AD^2 = (a + b Cos\theta)^2 + (bSin\theta)^2$

→$R^2 = a^2 + 2ab Cos\theta + b^2Cos^2 \theta + bSin^2 \theta$

→$R^2 = a^2 + 2ab Cos\theta + b^2 (Sin^2\theta + Cos^2 \theta )$

→$R= \sqrt{a^2 + b^2 + 2ab Cos \theta}$

Where R is $| \vec{A }+ \vec{B}|$.

• Which is also called resultant vector.

The angle subtended by resultant vector with $\vec{A}$ is :-

→$In \triangle{ADE}$

→$Tan\alpha = \dfrac{DE}{AE}$

→$Tan\alpha = \dfrac{b Sin\theta }{a+ b Cos \theta}$

Answer 2

Answer:

expression for magnitude and direction of their resultant