`A` and `B` are two vectors in a plane at an angle of `60^(@)` with each other. `C` is another vector perpendicular to the plane containing vector `A` and `B`. Which of the following relations is possible?
a) A + B = C
b) A + C= B
c) A X B = C
d) A x C = B
A
`A+B=C`
B
`A+C=B`
C
`AxxB=C`
D
`AxxC=B`
Text Solution
AI Generated Solution
The correct Answer is:
To solve the problem, we need to analyze the relationships between the vectors A, B, and C based on the information given.
### Step-by-Step Solution:
1. **Understanding the Vectors**:
- Vectors A and B are in a plane and form an angle of 60 degrees with each other.
- Vector C is perpendicular to the plane formed by A and B.
2. **Analyzing the Options**:
- **Option a: A + B = C**: This suggests that the sum of vectors A and B results in vector C. However, since A and B are in the same plane and C is perpendicular to that plane, this equation cannot hold true.
- **Option b: A + C = B**: This implies that adding vector C (which is perpendicular to the plane) to vector A gives vector B. Since A and C are not in the same plane, this relation is also not possible.
- **Option c: A × B = C**: The cross product of two vectors A and B results in a vector that is perpendicular to both A and B. Since C is defined as being perpendicular to the plane containing A and B, this relation is valid. Therefore, this option is possible.
- **Option d: A × C = B**: This suggests that the cross product of A and C results in vector B. However, since C is perpendicular to the plane of A and B, this relationship does not hold true as the cross product of a vector in the plane (A) and a vector perpendicular to that plane (C) will yield a vector in the plane, which cannot be equal to B.
3. **Conclusion**:
- The only valid relation among the given options is **Option c: A × B = C**.
### Final Answer:
The correct relation is **Option c: A × B = C**.
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