Question

Which inequality gives the next bound after \[|\vec{a}|^2+2|\vec{a}\cdot\vec{b}|+|\vec{b}|^2\] in the proof of the triangle inequality?

Options

• \[|\vec{a}|^2+2|\vec{a}+\vec{b}|+|\vec{b}|^2\]

• \[|\vec{a}|^2+2\vec{a}\cdot\vec{b}+|\vec{b}|^2\]

• \[|\vec{a}|^2+2|\vec{a}|\,|\vec{b}|+|\vec{b}|^2\]

• \[|\vec{a}|^2+2|\vec{a}|^2|\vec{b}|^2+|\vec{b}|^2\]

Answer 1

The bound \[|\vec{a}\cdot\vec{b}|\leq|\vec{a}|\,|\vec{b}|\] is applied to the middle term.
This yields \[|\vec{a}|^2+2|\vec{a}|\,|\vec{b}|+|\vec{b}|^2\].