The simplest operations on vectors are binary operations between vectors and scalar numbers. Binary operations are operators such as addition "+" or subtraction "-" in which the operator takes two inputs as in the case of "x+y". All binary operations between a vector and a scalar are accomplished by operating on each entry in the vector individually. The following shows vector-scalar addition as well as vector-scalar multiplication:
Since operations are defined in this way, the same properties that hold for scalar binary operations (like 1+1) also hold for vector-scalar binary operations. For instance, addition and multiplication are both commutative so if we swapped the order of either of the operations above we would obtain the same result. Formally we write that for a vector 
and scalar 
, the following are true: 
Subtraction and division are not commutative for the same reason they are not for scalars (
).
and scalar , the following are true: ). Excercise 2-1.
Perform the following operations:
One last thing to note is that vector-scalar multiplication by a positive constant changes the magnitude of a vector but does not change the direction as shown in Figure 2-1. Multiplying by a negative constant simply makes the vector run "anti-parallel" to the original direction.
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Figure 2-1.
Vector Scaling