Question

What is the linear velocity if the angular velocity vector is $\vec{\omega}=3 \hat{\imath}-4$ $\hat{\jmath}+\hat{k}$ and position vector is $\vec{r}=5 \hat{\imath}$ $-6 \hat{\jmath}+6 \hat{k} ?$

• A. $6 \hat{\imath}+2 \hat{\jmath}-3 \hat{k}$
• B. $-18 \, \hat{i}-13 \, \hat{j}+2 \, \hat{k}$
• C. $18 \, \hat{i}+13 \, \hat{j}-2 \, \hat{k}$
• D. $6 \, \hat{i}-2 \, \hat{j}+8 \, \hat{k}$

Answer 1

Answer: (B)

As we know that
$\vec{v}=\vec{\omega} \times \vec{r}=(3 \hat{i}-4 \hat{j}+\hat{k}) \times(5 \hat{i}-6 \hat{j}+6 \hat{k})$
$\therefore \vec{w} \times \vec{r}=\begin{vmatrix}\hat{\imath} & \hat{\jmath} & \hat{k} \\ 3 & -4 & 1 \\ 5 & -6 & 6\end{vmatrix}$
or $\vec{w} \times \vec{r}=\hat{i}(-24+6)+\hat{j}(5-18)+\hat{k}(-18+20)$
$=-18 \hat{i}-13 \hat{j}+2 \hat{k}$