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When vec(A).vec(B)= -|A| |B|, then...

When `vec(A).vec(B)= -|A| |B|`, then

A

`vecA` and `vecB` are perpendicular to each other

B

`vecA` and `vecB` act in the same direction

C

`vecA` and `vecB` act in the opposite direction

D

`vecA` and `vecB` can act in any direction

Text Solution

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The correct Answer is:
To solve the problem, we need to analyze the given condition and determine the angle between the vectors \( \vec{A} \) and \( \vec{B} \). ### Step-by-Step Solution: 1. **Understanding the Dot Product**: The dot product of two vectors \( \vec{A} \) and \( \vec{B} \) is given by the formula: \[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta \] where \( \theta \) is the angle between the two vectors. 2. **Given Condition**: According to the problem, we have: \[ \vec{A} \cdot \vec{B} = -|\vec{A}| |\vec{B}| \] 3. **Comparing the Two Expressions**: We can set the two expressions for the dot product equal to each other: \[ |\vec{A}| |\vec{B}| \cos \theta = -|\vec{A}| |\vec{B}| \] 4. **Dividing Both Sides**: Assuming \( |\vec{A}| \) and \( |\vec{B}| \) are not zero, we can divide both sides by \( |\vec{A}| |\vec{B}| \): \[ \cos \theta = -1 \] 5. **Finding the Angle**: The cosine of an angle is -1 at \( \theta = 180^\circ \). Therefore, we conclude: \[ \theta = 180^\circ \] 6. **Interpreting the Result**: An angle of 180 degrees indicates that the vectors \( \vec{A} \) and \( \vec{B} \) are pointing in opposite directions. 7. **Checking the Options**: - **Option A**: A and B are perpendicular to each other (90 degrees) - Incorrect. - **Option B**: A and B act in the same direction (0 degrees) - Incorrect. - **Option C**: A and B act in the opposite direction (180 degrees) - Correct. - **Option D**: Not specified, but not relevant since we found the correct answer. ### Final Answer: The correct option is **C: A and B act in the opposite direction**.
To solve the problem, we need to analyze the given condition and determine the angle between the vectors \( \vec{A} \) and \( \vec{B} \). ### Step-by-Step Solution: 1. **Understanding the Dot Product**: The dot product of two vectors \( \vec{A} \) and \( \vec{B} \) is given by the formula: \[ \vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta ...
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