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The resultant of vec(a) and vec(b) ma...

The resultant of ` vec(a)` and ` vec(b)` makes `alpha` with `vec(a)` and `beta` with `vec(b)`, then (a,b represent magnitudes of respective vectors) :

A

`apha lt beta`

B

`alphagt beta "if" Alt B`

C

`alpha lt beta "if" A=B`

D

`alpha lt beta "if" Alt B`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the relationship between the angles \( \alpha \) and \( \beta \) formed by the resultant vector with vectors \( \vec{a} \) and \( \vec{b} \) respectively. ### Step-by-Step Solution: 1. **Understanding the Vectors**: - Let \( \vec{a} \) and \( \vec{b} \) be two vectors with magnitudes \( a \) and \( b \) respectively. - The resultant vector \( \vec{R} \) is formed by the vector addition of \( \vec{a} \) and \( \vec{b} \). 2. **Angles with the Resultant**: - The angle \( \alpha \) is the angle between the resultant vector \( \vec{R} \) and vector \( \vec{a} \). - The angle \( \beta \) is the angle between the resultant vector \( \vec{R} \) and vector \( \vec{b} \). 3. **Relationship Between Angles and Magnitudes**: - The resultant vector is more inclined towards the vector with the greater magnitude. This means that if \( b > a \), the resultant vector \( \vec{R} \) will be closer to \( \vec{b} \) than to \( \vec{a} \). - Consequently, the angle \( \beta \) will be smaller than the angle \( \alpha \) when \( b > a \). This can be expressed as: \[ \beta < \alpha \quad \text{if } b > a \] 4. **Analyzing Different Cases**: - **Case 1**: If \( a < b \), then \( \beta < \alpha \). - **Case 2**: If \( a = b \), then \( \alpha = \beta \) because the resultant will be symmetrically inclined towards both vectors. - **Case 3**: If \( a > b \), then \( \alpha < \beta \). 5. **Conclusion**: - From the analysis, we can conclude that the angle \( \alpha \) is less than \( \beta \) when \( b > a \) and \( \alpha \) is greater than \( \beta \) when \( a > b \). Thus, the correct answer is: - \( \beta < \alpha \) if \( b > a \) - \( \alpha < \beta \) if \( a > b \) - \( \alpha = \beta \) if \( a = b \) ### Final Answer: - The correct option is: \( \alpha < \beta \) if \( a < b \).
To solve the problem, we need to analyze the relationship between the angles \( \alpha \) and \( \beta \) formed by the resultant vector with vectors \( \vec{a} \) and \( \vec{b} \) respectively. ### Step-by-Step Solution: 1. **Understanding the Vectors**: - Let \( \vec{a} \) and \( \vec{b} \) be two vectors with magnitudes \( a \) and \( b \) respectively. - The resultant vector \( \vec{R} \) is formed by the vector addition of \( \vec{a} \) and \( \vec{b} \). ...
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