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If vec(A)xxvec(B)=vec(C ), then which of...

If `vec(A)xxvec(B)=vec(C )`, then which of the following statements is wrong?

A

`vecCbotvecA`

B

`vecCbotvecB`

C

`vecCbot(vecA+vecB)`

D

`vecCbot(vecAxxvecB)`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to analyze the properties of the cross product of two vectors. Given that \( \vec{A} \times \vec{B} = \vec{C} \), we can derive the following conclusions about the relationships between the vectors involved. ### Step-by-Step Solution: 1. **Understanding the Cross Product**: The cross product of two vectors \( \vec{A} \) and \( \vec{B} \) results in a vector \( \vec{C} \) that is perpendicular to both \( \vec{A} \) and \( \vec{B} \). This is a fundamental property of the cross product. 2. **Analyzing the Statements**: We need to evaluate the correctness of the following statements: - **Statement 1**: \( \vec{C} \) is perpendicular to \( \vec{A} \). - **Statement 2**: \( \vec{C} \) is perpendicular to \( \vec{B} \). - **Statement 3**: \( \vec{C} \) is perpendicular to \( \vec{A} + \vec{B} \). - **Statement 4**: \( \vec{C} \) is perpendicular to \( \vec{A} \times \vec{B} \). 3. **Evaluating Each Statement**: - **Statement 1**: \( \vec{C} \) is perpendicular to \( \vec{A} \) — This is true because \( \vec{C} = \vec{A} \times \vec{B} \) is perpendicular to \( \vec{A} \). - **Statement 2**: \( \vec{C} \) is perpendicular to \( \vec{B} \) — This is also true because \( \vec{C} = \vec{A} \times \vec{B} \) is perpendicular to \( \vec{B} \). - **Statement 3**: \( \vec{C} \) is perpendicular to \( \vec{A} + \vec{B} \) — This statement is true as well. The vector \( \vec{A} + \vec{B} \) lies in the same plane as \( \vec{A} \) and \( \vec{B} \), and since \( \vec{C} \) is perpendicular to both, it is also perpendicular to their resultant. - **Statement 4**: \( \vec{C} \) is perpendicular to \( \vec{A} \times \vec{B} \) — This statement is incorrect. The vector \( \vec{C} \) is actually parallel to \( \vec{A} \times \vec{B} \) because \( \vec{C} = \vec{A} \times \vec{B} \). 4. **Conclusion**: The wrong statement among the options provided is Statement 4: \( \vec{C} \) is perpendicular to \( \vec{A} \times \vec{B} \). ### Final Answer: The wrong statement is: **\( \vec{C} \) is perpendicular to \( \vec{A} \times \vec{B} \)**.
To solve the problem, we need to analyze the properties of the cross product of two vectors. Given that \( \vec{A} \times \vec{B} = \vec{C} \), we can derive the following conclusions about the relationships between the vectors involved. ### Step-by-Step Solution: 1. **Understanding the Cross Product**: The cross product of two vectors \( \vec{A} \) and \( \vec{B} \) results in a vector \( \vec{C} \) that is perpendicular to both \( \vec{A} \) and \( \vec{B} \). This is a fundamental property of the cross product. 2. **Analyzing the Statements**: ...
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