The Question is from neet-chemistry , and it was created by the author P SRAVAN KUMAR

Question

The angular velocity of a rotating body is $\mathop \omega \limits^ 	o   = 4\,\widehat {m{i}}{m{ + }}\widehat {m{j}}{m{ - 2}}\widehat {m{k}}$. The linear velocity of the body whose position vector $2\widehat {m{i}}{m{ + 3}}\widehat {m{j}}{m{ - 3}}\widehat {m{k}}$ is

P SRAVAN KUMAR · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
	**Primary Concept**: Relationship between linear and angular velocity
	**Related Topics**: Vector cross product, rotational motion
	**Prerequisites**: Understanding of angular velocity, linear velocity, and vector cross product.
## SOLUTION PATHWAY
**Given Information**:
	Angular velocity: $\vec{\omega} = 4\hat{i} + \hat{j} - 2\hat{k}$
	Position vector: $\vec{r} = 2\hat{i} + 3\hat{j} - 3\hat{k}$
**Method & Steps**:
	1. Calculate the linear velocity using the formula $\vec{v} = \vec{\omega} 	imes \vec{r}$:
		Reasoning: The linear velocity of a point in a rotating body is given by the cross product of the angular velocity and the position vector.
		Formula/Rule: $\vec{v} = \vec{\omega} 	imes \vec{r}$
	2. Compute the cross product:
		Working: $\vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ 4 & 1 & -2 \ 2 & 3 & -3 \end{vmatrix} = \hat{i}((1)(-3) - (-2)(3)) - \hat{j}((4)(-3) - (-2)(2)) + \hat{k}((4)(3) - (1)(2))$
		Key Point: Remember the rules for calculating the determinant of a 3x3 matrix.
	3. Simplify:
		Result: $\vec{v} = \hat{i}(-3 + 6) - \hat{j}(-12 + 4) + \hat{k}(12 - 2) = 3\hat{i} + 8\hat{j} + 10\hat{k}$
		Verification: The resulting vector is the correct linear velocity.
 ## OPTION ANALYSIS
[A] :x: Incorrect calculation of the cross product.
[B] ✓: Correctly calculated linear velocity vector.
[C] :x: Incorrect calculation of the cross product.
[D] :x: Incorrect calculation of the cross product.
 ## LEARNING AIDS
	**Common Mistakes**:
		1.[Error 1]: Incorrectly calculating the cross product determinant. [How to avoid]: Practice calculating determinants and pay attention to signs.
		2.[Error 2]: Confusing the order of vectors in the cross product ($\vec{\omega} 	imes \vec{r}$ is not the same as $\vec{r} 	imes \vec{\omega}$). [How to avoid]: Always use the correct order as given in the formula.
	**Quick Tips**:
		1.[Helpful shortcut]: Use the determinant method to calculate the cross product.
		2.[Memory aid]: Remember the order of the vectors in the formula $\vec{v} = \vec{\omega} 	imes \vec{r}$.

P SRAVAN KUMAR · Answer

$5\,\widehat {\rm{i}}{\rm{ + 8}}\widehat {\rm{j}}{\rm{ + 14}}\widehat {\rm{k}}$

P SRAVAN KUMAR · Answer

The correct Option is "$3\,\widehat {\rm{i}}{\rm{ + 8}}\widehat {\rm{j}}{\rm{ + 10}}\widehat {\rm{k}}$"

P SRAVAN KUMAR · Answer

$8\,\widehat {\rm{i}} - {\rm{3}}\widehat {\rm{j}}{\rm{ + 2}}\widehat {\rm{k}}$

P SRAVAN KUMAR · Answer

$ - 8\,\widehat {\rm{i}}{\rm{ + 3}}\widehat {\rm{j}}{\rm{ + 2}}\widehat {\rm{k}}$

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