There are two ways of multiplying vectors which are of great importance in applications. The first of these is called the dot product, also called the scalar product and sometimes the inner product.
With this definition, there are several important properties satisfied by the dot product. In the statement of these properties, α and β will denote scalars and a,b,c will denote vectors.
You should verify these properties. Also be sure you understand that 3.4 follows from the first three and is therefore redundant. It is listed here for the sake of convenience.
Example 3.1.3 Find
This equals 0 + 2 + 0 + −3 = −1.
Example 3.1.4 Find the magnitude of a =
This is
The dot product satisfies a fundamental inequality known as the Cauchy Schwarz inequality. It has already been proved but here is another proof. This proof will be based only on the above axioms for the dot product.
Theorem 3.1.5 The dot product satisfies the inequality
 (3.6) 
Furthermore equality is obtained if and only if one of a or b is a scalar multiple of the other.
Proof: First note that if b = 0, both sides of 3.6 equal zero and so the inequality holds in this case. Indeed,

so a ⋅ 0 = 0. Therefore, it will be assumed in what follows that b≠0.
Define a function of t ∈ ℝ

Then by 3.2, f
f  = a ⋅ +
tb ⋅ 
= a ⋅ a + t +
tb ⋅ a + t^{2}b ⋅ b 

= ^{2} + 2
t +
^{2}
t^{2}. 
Now
f  = ^{2}

= ^{2}


= ^{2}
≥ 0 
for all t ∈ ℝ. In particular f
 (3.7) 
Multiplying both sides by

which yields 3.6.
From Theorem 3.1.5, equality holds in 3.6 whenever one of the vectors is a scalar multiple of the other. It only remains to verify this is the only way equality can occur. If either vector equals zero, then equality is obtained in 3.6 so it can be assumed both vectors are non zero and that equality is obtained in 3.7. This implies that f
You should note that the entire argument was based only on the properties of the dot product listed in 3.1  3.5. This means that whenever something satisfies these properties, the Cauchy Schwartz inequality holds. There are many other instances of these properties besides vectors in ℝ^{n}.
The Cauchy Schwartz inequality allows a proof of the triangle inequality for distances in ℝ^{n} in much the same way as the triangle inequality for the absolute value.
Theorem 3.1.6 (Triangle inequality) For a,b ∈ ℝ^{n}
 (3.8) 
and equality holds if and only if one of the vectors is a nonnegative scalar multiple of the other. Also
 (3.9) 
Proof: By properties of the dot product and the Cauchy Schwarz inequality,
^{2}
 =
⋅ =
+
+
+

= ^{2} + 2
+
^{2}
≤ ^{2} + 2
+
^{2}


≤ ^{2} + 2
+
^{2} =
^{2}
. 
Taking square roots of both sides you obtain 3.8.
It remains to consider when equality occurs. If either vector equals zero, then that vector equals zero times the other vector and the claim about when equality occurs is verified. Therefore, it can be assumed both vectors are nonzero. To get equality in the second inequality above, Theorem 3.1.5 implies one of the vectors must be a multiple of the other. Say b = αa. If α < 0 then equality cannot occur in the first inequality because in this case

Therefore, α ≥ 0.
To get the other form of the triangle inequality, a = a − b + b so

Therefore,
 (3.10) 
Similarly,
 (3.11) 
It follows from 3.10 and 3.11 that 3.9 holds. This is because