If `vec(A)xxvec(B)= vec(C )+vec(D)`, then select the correct alternative.
A
`B` is paralel to `C+D`
B
`A` is perpendicular to `C`
C
Component of `C` along `A=` component of `D` along `A`
D
Component of `C` along `A=`- component of `D` along `A`
Text Solution
AI Generated Solution
The correct Answer is:
To solve the problem, we start with the given equation involving vectors:
1. **Understanding the Equation**: The equation is given as:
\[
\vec{A} \times \vec{B} = \vec{C} + \vec{D}
\]
Here, \(\vec{A} \times \vec{B}\) represents the cross product of vectors \(\vec{A}\) and \(\vec{B}\).
2. **Properties of Cross Product**: One important property of the cross product is that the result is a vector that is perpendicular to both vectors involved in the product. Therefore, the vector \(\vec{A} \times \vec{B}\) is perpendicular to both \(\vec{A}\) and \(\vec{B}\).
3. **Analyzing the Right Side**: The right side of the equation, \(\vec{C} + \vec{D}\), is a vector sum. For the equation to hold true, the vector \(\vec{C} + \vec{D}\) must also be perpendicular to both \(\vec{A}\) and \(\vec{B}\).
4. **Conclusion about \(\vec{C}\)**: Since \(\vec{C} + \vec{D}\) is perpendicular to \(\vec{A}\), it implies that both \(\vec{C}\) and \(\vec{D}\) must be arranged in such a way that their resultant is perpendicular to \(\vec{A}\). However, the problem specifically asks about the relationship between \(\vec{A}\) and \(\vec{C}\).
5. **Final Statement**: Therefore, we can conclude that \(\vec{A}\) is perpendicular to \(\vec{C}\). This leads us to select the correct alternative that states:
**Option B**: \(\vec{A}\) is perpendicular to \(\vec{C}\).
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