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If vec(A) xx vec(B) = vec(B) xx vec(A), ...

If `vec(A) xx vec(B) = vec(B) xx vec(A)`, then the angle between `A and B` is

A

`(pi)/(2)`

B

`(pi)/(3)`

C

`pi`

D

`(pi)/(4)`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the condition given: \(\vec{A} \times \vec{B} = \vec{B} \times \vec{A}\). ### Step-by-Step Solution: 1. **Understanding the Cross Product**: The cross product of two vectors \(\vec{A}\) and \(\vec{B}\) is given by: \[ \vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin(\theta) \hat{n} \] where \(\theta\) is the angle between the two vectors, and \(\hat{n}\) is the unit vector perpendicular to the plane formed by \(\vec{A}\) and \(\vec{B}\). 2. **Properties of the Cross Product**: The cross product has the property that: \[ \vec{B} \times \vec{A} = -(\vec{A} \times \vec{B}) \] This means that \(\vec{B} \times \vec{A}\) is equal in magnitude but opposite in direction to \(\vec{A} \times \vec{B}\). 3. **Setting the Condition**: Given that \(\vec{A} \times \vec{B} = \vec{B} \times \vec{A}\), we can substitute the property of the cross product: \[ \vec{A} \times \vec{B} = -(\vec{A} \times \vec{B}) \] This implies: \[ \vec{A} \times \vec{B} = \vec{0} \] 4. **Condition for Zero Cross Product**: The cross product \(\vec{A} \times \vec{B}\) is zero if: - Either \(|\vec{A}| = 0\) (meaning \(\vec{A}\) is the zero vector), - Or \(|\vec{B}| = 0\) (meaning \(\vec{B}\) is the zero vector), - Or \(\sin(\theta) = 0\) (which occurs when \(\theta = 0\) or \(\theta = \pi\)). 5. **Analyzing the Angles**: Since we are looking for the angle between \(\vec{A}\) and \(\vec{B}\), we discard the cases where either vector is the zero vector. Thus, we focus on: \[ \sin(\theta) = 0 \implies \theta = 0 \text{ or } \theta = \pi \] 6. **Conclusion**: In the context of angles between vectors, \(\theta = 0\) means the vectors are in the same direction, and \(\theta = \pi\) means the vectors are in opposite directions. The question does not specify that both vectors are non-zero, so we conclude: \[ \text{The angle between } \vec{A} \text{ and } \vec{B} \text{ is } \pi \text{ radians (or 180 degrees)}. \] ### Final Answer: The angle between \(\vec{A}\) and \(\vec{B}\) is \(\pi\) radians.
To solve the problem, we need to analyze the condition given: \(\vec{A} \times \vec{B} = \vec{B} \times \vec{A}\). ### Step-by-Step Solution: 1. **Understanding the Cross Product**: The cross product of two vectors \(\vec{A}\) and \(\vec{B}\) is given by: \[ \vec{A} \times \vec{B} = |\vec{A}| |\vec{B}| \sin(\theta) \hat{n} ...
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