Vectors III:

We always try to represent vectors in terms of their unit vector components. This makes all vector operations easy (well, at least easier).

Besides to addition and subtraction you can multiply vectors - in three ways (but we never divide vectors). The first is multiplication by a scalar – it makes the length of the vector change, but not its orientation (except if the scalar is negative – then it reverses the direction). 

For example,  (If the multiplier were -3, each term in the result would be negative).

We define the dot product of two vectors as

,

where 2 is the angle between A and B. The result is a scalar (which is why this operation is also called the scalar product and, sometimes, the inner product). We make this definition (as always) because it is useful to do so; i.e. it helps us use vectors to represent physical situations.  Here’s an example: As we always use unit vectors, take the dot product of  and  To do this, take the dot product of each term – this results in 9 terms in the solution.  Writing this out in all its detail we have

 

The first term is  and the second is  Now  and  Similar results hold for the other terms. The dot product of these two vectors then becomes  (a scalar, by definition).

Moral: to take the dot product of two vectors, take the scalar product of like components and add them up – the result is always a number (i.e. a scalar).                        

We define the cross product of two vectors as

The result is defined to be a vector (which is why this operation is also called the vector product and, sometimes, the outer product).