Question:
What is the linear velocity if angular velocity $\vec{\omega} = 3\hat{i} - 4\hat{j} + \hat{k}$ and radius $\vec{r} = (5\hat{i} - 6\hat{j} + 6\hat{k})$ ?
What is the linear velocity if angular velocity $\vec{\omega} = 3\hat{i} - 4\hat{j} + \hat{k}$ and radius $\vec{r} = (5\hat{i} - 6\hat{j} + 6\hat{k})$ ?
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Always use the order $\vec{\omega} \times \vec{r}$. Reversing the order changes the sign of the result.
Updated On: Apr 30, 2026
- $(-30\hat{i} - 13\hat{j} - 38\hat{k})$
- $(8\hat{i} - 10\hat{j} + 7\hat{k})$
- $(-18\hat{i} - 13\hat{j} + 2\hat{k})$
- $(-2\hat{i} - 2\hat{j} - 5\hat{k})$
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The Correct Option is C
Solution and Explanation
Step 1: Formula
\[ \vec{v} = \vec{\omega} \times \vec{r} \]
Step 2: Cross Product Calculation
\[ \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & -4 & 1 \\ 5 & -6 & 6 \end{vmatrix} \]
\[ \vec{v} = \hat{i}(-24 - (-6)) - \hat{j}(18 - 5) + \hat{k}(-18 - (-20)) \]
Step 3: Simplifying
\[ \vec{v} = -18\hat{i} - 13\hat{j} + 2\hat{k} \]
Step 4: Conclusion
The linear velocity vector is \( -18\hat{i} - 13\hat{j} + 2\hat{k} \).
Final Answer: (C)
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