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What is the value of (vec(A) + vec(B)) ....

What is the value of `(vec(A) + vec(B)) .(vec(A) xx vec(B))` ?

A

0

B

`A^(2) - B^(2)`

C

`A^(2) + B^(2) + 2AB`

D

none of these

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we need to evaluate the expression \((\vec{A} + \vec{B}) \cdot (\vec{A} \times \vec{B})\). ### Step-by-Step Solution: 1. **Understanding the Vectors**: - We have two vectors \(\vec{A}\) and \(\vec{B}\). - The expression involves both the sum of these vectors and their cross product. 2. **Properties of Cross Product**: - The cross product \(\vec{A} \times \vec{B}\) results in a vector that is perpendicular to the plane formed by \(\vec{A}\) and \(\vec{B}\). 3. **Sum of Vectors**: - The vector \(\vec{A} + \vec{B}\) lies in the same plane as \(\vec{A}\) and \(\vec{B}\). 4. **Angle Between Vectors**: - Since \(\vec{A} + \vec{B}\) lies in the plane formed by \(\vec{A}\) and \(\vec{B}\), and \(\vec{A} \times \vec{B}\) is perpendicular to this plane, the angle \(\theta\) between \((\vec{A} + \vec{B})\) and \((\vec{A} \times \vec{B})\) is \(90^\circ\). 5. **Dot Product Calculation**: - The dot product of two vectors is given by the formula: \[ \vec{X} \cdot \vec{Y} = |\vec{X}| |\vec{Y}| \cos(\theta) \] - Here, \(\theta = 90^\circ\), and \(\cos(90^\circ) = 0\). 6. **Final Result**: - Therefore, the dot product \((\vec{A} + \vec{B}) \cdot (\vec{A} \times \vec{B})\) is: \[ (\vec{A} + \vec{B}) \cdot (\vec{A} \times \vec{B}) = 0 \] ### Conclusion: The value of \((\vec{A} + \vec{B}) \cdot (\vec{A} \times \vec{B})\) is \(0\). ---
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