Answer:
Use the Cauchy-Schwarz inequality
|v · w| ≤ ||v|| · ||w||
to prove the Triangle Inequality
||v + w|| ≤ ||v|| + ||w||.
(Hint: Begin by computing ||v + w||2 with dot products, and then plug in the
Cauchy-Schwarz inequality when the opportunity arises.)
Solution.
||v + w||2 = (v + w) · (v + w)
= v · v + v · w + w · v + w · w
= ||v||2 + 2(v · w) + ||w||2
≤ ||v||2 + 2||v||||w|| + ||w||2 by the C.S. Inequality
= (||v|| + ||w||)
2
Taking the square root of both sides, we conclude that
||v + w|| ≤ ||v|| + ||w||Step-by-step explanation:
