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.
a = 

b = 

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.
a = 

b = 

Then,
C = ab' = 

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.
a'b = 0 1 * 2 3
a'b = 0*2 + 1*3 = 3 The term bc' is an outer product. Here, it is a 2 x 3 matrix, as shown below.
bc' = 2 3 * 4 5 6
bc' = 2*4 2*5 2*6 3*4 3*5 3*6 = 8 10 12 12 15 18
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.