Question

Consider vectors A and B. Show that |A + B| ≤ |A| + |B|

Answer 1

The inequality |A + B| ≤ |A| + |B|, known as the Triangle Inequality, can be proven by squaring both sides and using the Cauchy-Schwarz inequality to confirm the left side is less than or equal to the right side.

Step-by-step explanation:

The inequality |A + B| ≤ |A| + |B| is known as the Triangle Inequality in the context of vectors. It's named so because the lengths of any two sides of a triangle are always greater than or equal to the length of the third side.

Here's how we can prove it:

1. First, we square both sides of the inequality, which gives us: |A + B|^2 ≤ (|A| + |B|)^2
2. Then we calculate the left and right hand sides:
   1. Left side: |A + B|^2 = A^2 + 2A.B + B^2
   2. Right side: (|A| + |B|)^2 = A^2 + 2|A||B|+ B^2
3. Due to the Cauchy-Schwarz inequality, we know that 2A.B ≤ 2|A||B|. So we have that the left side is less than or equal to the right side, which proves our original inequality.

Learn more about Triangle Inequality