Question

Using commutativity of the scalar product, which expression equals \[|\vec{a}+\vec{b}|^2\]?

Options

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

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

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

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

Answer 1

Since \[\vec{a}\cdot\vec{b}=\vec{b}\cdot\vec{a}\], the two middle terms combine.
This gives \[|\vec{a}+\vec{b}|^2=|\vec{a}|^2+2\vec{a}\cdot\vec{b}+|\vec{b}|^2\].