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If vec(A)+vec(B)+vec(C )=0 then vec(A)xx...

If `vec(A)+vec(B)+vec(C )=0` then `vec(A)xx vec(B)` is

A

`vec(C )xx vec(B)`

B

`vec(B)xx vec(C )`

C

`vec(A)xx vec(C )`

D

Zero

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we start with the equation given: 1. **Given Equation**: \[ \vec{A} + \vec{B} + \vec{C} = 0 \] 2. **Rearranging the Equation**: From the equation above, we can express one vector in terms of the others. Let's express \(\vec{C}\): \[ \vec{C} = -(\vec{A} + \vec{B}) \] 3. **Cross Product with \(\vec{B}\)**: We need to find \(\vec{A} \times \vec{B}\). We can use the expression for \(\vec{C}\) in our calculations: \[ \vec{A} + \vec{B} + \vec{C} = 0 \implies \vec{A} + \vec{B} = -\vec{C} \] 4. **Cross Multiplying**: Now, we can take the cross product of both sides with \(\vec{B}\): \[ (\vec{A} + \vec{B}) \times \vec{B} = -\vec{C} \times \vec{B} \] 5. **Expanding the Left Side**: Using the distributive property of the cross product: \[ \vec{A} \times \vec{B} + \vec{B} \times \vec{B} = -\vec{C} \times \vec{B} \] 6. **Simplifying**: Since the cross product of any vector with itself is zero: \[ \vec{B} \times \vec{B} = \vec{0} \] Therefore, we have: \[ \vec{A} \times \vec{B} = -\vec{C} \times \vec{B} \] 7. **Using the Property of Cross Product**: The negative of the cross product can be rewritten: \[ -\vec{C} \times \vec{B} = \vec{B} \times \vec{C} \] 8. **Final Result**: Thus, we find: \[ \vec{A} \times \vec{B} = \vec{B} \times \vec{C} \] So, the final answer is: \[ \vec{A} \times \vec{B} = \vec{B} \times \vec{C} \]
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