Question

two vectors A and B makes angle alpha and beta respectively with their resultant vector then magnitude of the resultant vector is ​

Answer 1

$\left |\vec{c} \right |=\left | \vec{a} \right |\left ( sin\alpha cot\beta + cos\alpha \right )$

Explanation:

• Here, we observe that $\vec{c}$ represents the resultant of$\vec{a}$ and $\vec{b}$. (refer the figure)

• a, b, c are represents the magnitude of vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ respectively.

By applying the sine rule, we can write as;

• $\dfrac{a}{sin\beta} = \dfrac{b}{sin\alpha} = \dfrac{c}{sin(\pi - \ ( \alpha + \beta)}$

Now, from first and third proportion, we write as;

• $\dfrac{a}{sin\beta} = \dfrac{c}{sin(\pi - \ ( \alpha + \beta)}$

• $c = a \dfrac{(sin\alpha cos\beta + cos\alpha sin\beta)}{sin\beta}$

• $c = a \left ( sin\alpha \dfrac{cos\beta}{sin\beta} + cos\alpha \right )$

• $\left |\vec{c}\right |=\left | \vec{a} \right |\left ( sin\alpha \dfrac{cos\beta}{sin\beta} + cos\alpha \right )$

• $\left |\vec{c}\right |=\left | \vec{a} \right |\left ( sin\alpha cot\beta + cos\alpha \right )$

This is the magnitude of resultant vector.