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If vec(B) = n vec(A) and vec(A) is anti ...

If `vec(B) = n vec(A) and vec(A)` is anti parallel with `vec(B)`, then n is:

A

a vector

B

negative and dimension less

C

a scalar

D

a negative scalar

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The correct Answer is:
To solve the problem, we need to analyze the relationship between the vectors \(\vec{A}\) and \(\vec{B}\) given that \(\vec{B} = n \vec{A}\) and that \(\vec{A}\) is anti-parallel to \(\vec{B}\). ### Step-by-Step Solution: 1. **Understanding Anti-parallel Vectors**: - Two vectors are said to be anti-parallel if they point in opposite directions. This means that if \(\vec{A}\) points in a certain direction, \(\vec{B}\) must point in the exact opposite direction. 2. **Given Relationship**: - We are given that \(\vec{B} = n \vec{A}\). This means that \(\vec{B}\) is a scalar multiple of \(\vec{A}\). 3. **Analyzing the Scalar \(n\)**: - If \(n\) is positive, then \(\vec{B}\) would point in the same direction as \(\vec{A}\), which contradicts the condition that they are anti-parallel. - Therefore, \(n\) must be negative for \(\vec{B}\) to point in the opposite direction of \(\vec{A}\). 4. **Determining the Nature of \(n\)**: - Since \(n\) is a scalar that relates two vectors, it cannot be a vector itself. Hence, \(n\) must be a scalar quantity. - Additionally, since we established that \(n\) is negative, we can conclude that \(n\) is a negative scalar. 5. **Dimensions of \(n\)**: - The dimensions of \(\vec{B}\) and \(\vec{A}\) must be the same because they are vectors. Since \(\vec{B} = n \vec{A}\), the scalar \(n\) must be dimensionless to ensure that the dimensions on both sides of the equation match. 6. **Final Conclusion**: - Therefore, \(n\) is a negative scalar and dimensionless. ### Answer: The value of \(n\) is a negative scalar and dimensionless.
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