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If vec(A), vec(B) and vec(C ) are three ...

If `vec(A), vec(B)` and `vec(C )` are three vectors, then the wrong relation is :

A

`vec(A) + (vec(B) + vec(C )) = (vec(A) + vec(B)) + vec(C )`

B

`vec(A).(vec(B) + vec(C )) = (vec(A).vec(B)) + vec(A).vec(C )`

C

`vec(A) xx (vec(B) + vec(C )) = vec(A) xx vec(B) + vec(A) xx vec(C )`

D

`(vec(A) xx vec(B)) . vec(C ) = vec(A) xx vec(B). vec(C )`

Text Solution

AI Generated Solution

The correct Answer is:
To determine the wrong relation among the given options involving vectors \( \vec{A}, \vec{B}, \) and \( \vec{C} \), we will analyze the properties of vector operations, specifically focusing on addition, dot product, and cross product. ### Step-by-Step Solution: 1. **Understanding Vector Addition**: - Vectors can be added together using the associative and commutative properties. - For any vectors \( \vec{A}, \vec{B}, \vec{C} \): - \( \vec{A} + \vec{B} = \vec{B} + \vec{A} \) (Commutative Property) - \( \vec{A} + (\vec{B} + \vec{C}) = (\vec{A} + \vec{B}) + \vec{C} \) (Associative Property) **Hint**: Check if the properties of addition hold true for the vectors involved. 2. **Understanding Dot Product**: - The dot product of two vectors results in a scalar. - For any vectors \( \vec{A}, \vec{B}, \vec{C} \): - \( \vec{A} \cdot \vec{B} = \vec{B} \cdot \vec{A} \) (Commutative Property) - \( \vec{A} \cdot (\vec{B} + \vec{C}) = \vec{A} \cdot \vec{B} + \vec{A} \cdot \vec{C} \) (Distributive Property) **Hint**: Verify if the dot product maintains its properties when applied to combinations of vectors. 3. **Understanding Cross Product**: - The cross product of two vectors results in another vector. - For any vectors \( \vec{A}, \vec{B}, \vec{C} \): - \( \vec{A} \times \vec{B} = -(\vec{B} \times \vec{A}) \) (Anticommutative Property) - \( \vec{A} \times (\vec{B} + \vec{C}) = \vec{A} \times \vec{B} + \vec{A} \times \vec{C} \) (Distributive Property) **Hint**: Remember that the cross product is not commutative and results in a vector. 4. **Identifying the Wrong Relation**: - The question states that we need to find the wrong relation among the options. - One of the relations could involve a combination of cross product and dot product, such as \( \vec{A} \times (\vec{B} \cdot \vec{C}) \). - Since \( \vec{B} \cdot \vec{C} \) is a scalar, the expression \( \vec{A} \times (\vec{B} \cdot \vec{C}) \) is not defined because the cross product requires two vectors, not a vector and a scalar. **Conclusion**: The wrong relation is \( \vec{A} \times (\vec{B} \cdot \vec{C}) \). ### Final Answer: The wrong relation is \( \vec{A} \times (\vec{B} \cdot \vec{C}) \).
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