Look at the first triangle inequality:

||y + z|| ≤ ||y|| + ||z||

We can illustrate this by the following diagram.

The inequality illustrates the fact that the length of the longest edge of a triangle is less than the sum of the lengths of the other sides. (Equality occurs when y and z are parallel vectors, so that the triangle collapses to a line.)

The second triangle inequality

||yz|| ≤ ||yw|| + ||wz||

is similarly illustrated by the following diagram.

Just close this window to return to the lecture.