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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 given vector equation: \[ \vec{A} + \vec{B} + \vec{C} = 0 \] From this equation, we can express one of the vectors in terms of the others. Let's express \(\vec{C}\) in terms of \(\vec{A}\) and \(\vec{B}\): \[ \vec{C} = -(\vec{A} + \vec{B}) \] Now, we need to find the value of \(\vec{A} \times \vec{B}\). We can use the relation we derived for \(\vec{C}\): \[ \vec{A} \times \vec{B} = \vec{A} \times (-\vec{C}) \quad \text{(since } \vec{C} = -(\vec{A} + \vec{B})\text{)} \] Using the property of the cross product, we know that: \[ \vec{A} \times (-\vec{C}) = -(\vec{A} \times \vec{C}) \] Next, we can also express \(\vec{A} \times \vec{C}\) using \(\vec{C} = -(\vec{A} + \vec{B})\): \[ \vec{A} \times \vec{C} = \vec{A} \times -(\vec{A} + \vec{B}) = -(\vec{A} \times \vec{A} + \vec{A} \times \vec{B}) \] Since the cross product of any vector with itself is zero, we have: \[ \vec{A} \times \vec{A} = 0 \] Thus, we simplify: \[ \vec{A} \times \vec{C} = -(\vec{0} + \vec{A} \times \vec{B}) = -\vec{A} \times \vec{B} \] Now substituting back, we find: \[ \vec{A} \times \vec{B} = -(-\vec{A} \times \vec{B}) = \vec{A} \times \vec{B} \] This means that: \[ \vec{A} \times \vec{B} = \vec{B} \times \vec{C} \] Finally, we can conclude that: \[ \vec{A} \times \vec{B} = \vec{B} \times \vec{C} \] Thus, the value of \(\vec{A} \times \vec{B}\) is: \[ \vec{B} \times \vec{C} \] ### Final Answer: \[ \vec{A} \times \vec{B} = \vec{B} \times \vec{C} \] ---
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