The Question is from neet-physics motion-in-a-plane , and it was created by the author ABDUL JILANI BASHA

Question

The component of $\overrightarrow A $ along $\overrightarrow B $  is 2 times that of the component of $\overrightarrow B $  along $\overrightarrow A $. Then A : B is

ABDUL JILANI BASHA · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
	**Primary Concept**: Vector Projection and Dot Product
	**Related Topics**:  Components of vectors, magnitudes of vectors, ratios
	**Prerequisites**: Understanding of vector projection, dot product formula, and basic algebra.
## SOLUTION PATHWAY
**Given Information**:
	* The component of $\overrightarrow{A}$ along $\overrightarrow{B}$ is twice the component of $\overrightarrow{B}$ along $\overrightarrow{A}$.
**Method & Steps**:
	1. Express the component of $\overrightarrow{A}$ along $\overrightarrow{B}$ using the dot product:
		Reasoning: The component of $\overrightarrow{A}$ along $\overrightarrow{B}$ is given by $\frac{\overrightarrow{A} \cdot \overrightarrow{B}}{|B|}$
		Formula/Rule:  $\frac{\overrightarrow{A} \cdot \overrightarrow{B}}{|B|}$
	2. Express the component of $\overrightarrow{B}$ along $\overrightarrow{A}$ using the dot product:
		Working: The component of $\overrightarrow{B}$ along $\overrightarrow{A}$ is given by $\frac{\overrightarrow{A} \cdot \overrightarrow{B}}{|A|}$
		Key Point: The dot product is commutative, i.e., $\overrightarrow{A} \cdot \overrightarrow{B} = \overrightarrow{B} \cdot \overrightarrow{A}$
	3. Set up the equation based on the given condition:
		Working:  Given that the component of $\overrightarrow{A}$ along $\overrightarrow{B}$ is twice the component of $\overrightarrow{B}$ along $\overrightarrow{A}$, we have: $\frac{\overrightarrow{A} \cdot \overrightarrow{B}}{|B|} = 2 \frac{\overrightarrow{A} \cdot \overrightarrow{B}}{|A|}$
	4. Simplify and find the ratio of A:B:
		Working: $\frac{1}{|B|} = \frac{2}{|A|}$. Therefore, $|A| = 2|B|$. This implies $\frac{|A|}{|B|} = \frac{2}{1}$
		Result: A : B = 2 : 1
		Verification: The ratio matches the correct answer
## OPTION ANALYSIS
[A] :x: 1:4 is incorrect because the ratio is inverted and not equal to the calculated ratio of 2:1.
[B] ✓: 2:1 is correct as it matches the calculated ratio.
[C] :x: $\sqrt{3}:2$ is incorrect because it does not match the calculated ratio.
[D] :x: 6:1 is incorrect because it does not match the calculated ratio.
## LEARNING AIDS
	**Common Mistakes**:
		1. **Incorrectly using the dot product**: Mistaking the order of vectors in dot product and the component formula.
		2. **Inverting the ratio**: Getting the ratio of B:A instead of A:B.
	**Quick Tips**:
		1. Always remember the dot product is commutative $\overrightarrow{A} \cdot \overrightarrow{B} = \overrightarrow{B} \cdot \overrightarrow{A}$
		2. The component of a vector along another is always the projection of the first vector onto the second.

ABDUL JILANI BASHA · Answer

$1{\rm{ }}:{\rm{ }}4$

ABDUL JILANI BASHA · Answer

The correct Option is "$2{\rm{ }}:{\rm{ }}1$"

ABDUL JILANI BASHA · Answer

$\sqrt 3 :2$

ABDUL JILANI BASHA · Answer

$6{\rm{ }}:{\rm{ }}1$

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