Proof of Triangle Inequality and Equality Condition - SEMATH INFO -

  For any real vectors $\mathbf{a}$ and $\mathbf{b}$, holds. This inequality is called triangle inequality . The proof is below.

Proof

  Geometrically, the triangular inequality is an inequality expressing that the sum of the lengths of two sides of a triangle is longer than the length of the other side as shown in the figure below. The proof is as follows.
  Let $\mathbf{a}$ and $\mathbf{b}$ be real vectors. The square of the norm of $\mathbf{a} + \mathbf{b}$ satisfies , where we used that the inner product between two vectors is symmetric, , and $ \left(\mathbf{a}, \mathbf{b}\right) \leq | \left(\mathbf{a}, \mathbf{b}\right)| $ generally holds.
  Applying the Schwarz inequality, , to this inequality, we see that . From this and , we see that the triangle inequality holds.   We will discuss the equality condition of the triangular inequality.
  First, one of the equality condition of the triangle inequality is $ \mathbf{a} = 0 $ or $ \mathbf{b} = 0 $ , because, in this case, it is clear that . In other cases, that is, if $ \mathbf{a} \neq 0 $ and $ \mathbf{b} \neq 0 $ , the equality holds if and only if $\mathbf{a}$ and $\mathbf{b}$ are parallel and in the same direction, , where $t > 0$.
  This can be understood by considering as follows. In the above proof of the triangle inequality, we only use relations that hold for all vectors other than the Schwarz inequality, , and the inequality, . Therefore, the equality of the triangular inequality holds if and only if the equality of $(1)$ and that of $(2)$ hold. We thus consider the equality condition of $(1)$ and $(2)$.
  It is known that the equality condtion of Schwarz inequality $(1)$ is that $\mathbf{a}$ and $\mathbf{b}$ are parallel, . Here $t$ can be a posive value or a negative value. If $t$ is positive, the equality of $(2)$ holds because . On the other hand, if $t$ is negative, the equality of $(2)$ does not hold because . Therefore, the equality of $(2)$ holds only if $t$ is positive.
  We thus see that the equality condtion of $(1)$ and $(2)$ is that , where $t>0$. This is the equality condition of the triangle inequality.