What is the angle between vector 2A and vector -3A (negative 3A) when they are drawn from a common origin?

The angle between the two vectors \( 2\mathbf{A} \) and \( -3\mathbf{A} \) when drawn from a common origin is \( 180^\circ \).

This is because the vectors are scalar multiples of each other, but one is a positive multiple (\( 2\mathbf{A} \)), and the other is a negative multiple (\( -3\mathbf{A} \)). Vectors that are scalar multiples of each other but with opposite signs point in exactly opposite directions, creating an angle of \( 180^\circ \) between them.

Would you like a detailed explanation or have any other questions?

Here are five related questions you might find interesting:
1. What is the angle between two vectors that are scalar multiples with the same sign?
2. How do you calculate the angle between two arbitrary vectors using their dot product?
3. What is the magnitude of the vector \( 2\mathbf{A} - 3\mathbf{A} \)?
4. How do you determine if two vectors are perpendicular using their dot product?
5. What is the geometric interpretation of the dot product between two vectors?

**Tip:** Remember, if two vectors are parallel and point in the same direction, the angle between them is \( 0^\circ \). If they point in opposite directions, the angle is \( 180^\circ \).

Explore the concept of finding the angle between two vectors when one is a positive scalar multiple and the other is a negative scalar multiple. This problem demonstrates why the angle between vector 2A and vector -3A is 180°, using principles from vector algebra. Ideal for students in Grades 10-12.

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