How do you multiply vectors?

Vectors can be multiplied using either the dot product (scalar product) or the cross product (vector product).

In physics, vectors are quantities that have both magnitude and direction. When multiplying vectors, there are two methods that can be used: the dot product and the cross product. The method you choose depends on what you're trying to achieve.

The dot product, also known as the scalar product, is used when you want to find the magnitude of one vector in the direction of another. The dot product of two vectors A and B is calculated as A.B = |A||B|cosθ, where |A| and |B| are the magnitudes of the vectors and θ is the angle between them. The result is a scalar quantity, meaning it has magnitude but no direction.

On the other hand, the cross product, also known as the vector product, is used when you want to find a vector that is perpendicular to the plane containing the two original vectors. The cross product of two vectors A and B is calculated as AxB = |A||B|sinθn, where n is a unit vector perpendicular to the plane containing A and B. The direction of n is determined by the right-hand rule. The result is a vector quantity, meaning it has both magnitude and direction.

It's important to note that vector multiplication is not commutative. This means that A.B is not necessarily equal to B.A, and AxB is not necessarily equal to BxA. In fact, while A.B is always equal to B.A, AxB is equal to -BxA. This is a key difference between vector multiplication and scalar multiplication, and it's something you need to keep in mind when working with vectors.

In summary, when multiplying vectors, you can use either the dot product or the cross product, depending on whether you want a scalar or a vector result. The dot product gives you the magnitude of one vector in the direction of another, while the cross product gives you a vector that is perpendicular to the plane containing the two original vectors.