Vector Multiplication
The multiplication of a vector by a vector produces some interesting results,
known as the vector inner product and as the vector outer product.
Prerequisite: This material assumes familiarity with
matrix multiplication.
Vector Inner Product
Assume that a and b are
vectors,
each with the same number of elements. Then, the
inner product of a and
b is s.
a'b
= b'a = s
where
a and b are column vectors,
each having n elements,
a' is the transpose of a, which makes
a' a row vector,
b' is the transpose of b, which makes
b' a row vector, and
s is a scalar; that is, s is a real number  not a matrix.
Note this interesting result. The product of two matrices is usually
another matrix. However, the inner product of two vectors is different.
It results in a real number  not a matrix. This is illustrated below.
Then,
a'b = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32
Thus, the inner product of a'b is
equal to 32.
Note: The inner product is also known as the
dot product or as the scalar product.
Vector Outer Product
Assume that a and b are
vectors. Then, the
outer product of a and
b is C.
ab'= C
where
a is a column vector, having m elements,
b is a column vector, having n elements,
b' is the transpose of b, which makes
b' a row vector, and
C is a rectangular m x n matrix
Unlike the inner product, the outer product of two vectors produces
a rectangular matrix, not a scalar. This is illustrated below.
Then,
C = ab' = 

v * x 
v * y 
v * z 

w * x 
w * y 
w * z 

Notice that the elements of Matrix C consist of
the product of elements from Vector A crossed with
elements from Vector B. Thus, Matrix C
winds up being a matrix of cross products from the two vectors.
Test Your Understanding
Problems
Consider the vectors shown below  a, b,
and c
Using a, b, and c,
answer the questions below.
1. Find a'b,
the inner product of a and b.
2. Find bc',
the outer product of b and c.
3. True or false:
bc' = cb'
Solutions
The term a'b is an inner product,
which is equal to 3. The solution appears below.
The term bc' is an outer product. Here, it is
a 2 x 3 matrix, as shown below.
The statement bc'
= cb' is false.
Note that b is a 2 x 1 vector and c
is a 3 x 1 vector. Therefore, bc'
is a 2 x 3 matrix, and cb' is a
3 x 2 matrix. Because bc' and
cb' have different
dimensions,
they cannot be equal.