Question

Velocity and acceleration of a particle at some instant are $v=(3\hat{i}-4\hat{j}+2\hat{k})m/s$ and $a=(2\hat{i}+\hat{j}-2\hat{k})m/s^2$. What is the angle between $v$ and $a$ - acute, obtuse or ${90}^o$?

Answer 1

We know $\cos \theta$ =$\frac{v.a}{v\times a}$-----(1)

$v.a=-2$

$|v|=\sqrt{3^2+(-4{)}^2+(2{)}^2}=\sqrt{29}|a|=\sqrt{2^2+(1{)}^2+(-2{)}^2}=3$

Putting the found values in (1), we get

$cos\theta =\frac{v.a}{|v|\times |a|}=\frac{-2}{3\sqrt{9}}\theta ={cos}^{-1}\left(\frac{-2}{3\sqrt{9}}\right)=97.11$

Thus the angle between $v$ and $a$ is obtuse .