Question

The linear velocity of a rotating body is given by
$\overrightarrow{v}=\overrightarrow{\omega }\times \overrightarrow{r}$, where $\overrightarrow{\omega }$ is the angular velocity and $\overrightarrow{r}$ is the radius vector. The angular velocity of a body is $\overrightarrow{\omega }=\hat{i}-2\hat{j}+2\hat{k}$ and the radius vector $\overrightarrow{r}=4\hat{j}-3\hat{k}$ , then $|\overrightarrow{v}|$ is $\sqrt{x}\text{units}$. Find $x$.

Answer 1

Given,
Angular velocity, $\overrightarrow{\omega }=\hat{i}-2\hat{j}+2\hat{k}$
radius vector, $\overrightarrow{r}=4\hat{j}-3\hat{k}$
$\overrightarrow{v}=\overrightarrow{\omega }\times \overrightarrow{r}$
$\overrightarrow{v}=\overrightarrow{\omega }\times \overrightarrow{r}=\left|\begin{array}{ccc}\hat{i}& \hat{j}& \hat{k}\\ 1& -2& 2\\ 0& 4& -3\end{array}\right|$
$=\hat{i}(6-8)-\hat{j}(-3)+4\hat{k}$
$=-2\hat{i}+3\hat{j}+4\hat{k}$
$|\overrightarrow{v}|=\sqrt{(-2{)}^2+(3{)}^2+4^2})=\sqrt{29}\text{units}$
Final Answer: $\left(29\right)$