For any vectors $\vec { u }$ and $\vec { v }$ , $( \vec { u } + \vec { V } ) \cdot ( \vec { u } - \vec { v } ) = \| \vec { u } \| ^ { 2 } - \| \vec { v } \| ^ { 2 }$ .

Question

Quizplus · Accepted Answer

Expanding the left side using the distributive property of dot product gives $\vec{u} \cdot \vec{u} + \vec{u} \cdot (-\vec{v}) + \vec{v} \cdot \vec{u} - \vec{v} \cdot \vec{v}$. Simplifying, we get $\|\vec{u}\|^2 - \|\vec{v}\|^2$, since $\vec{u} \cdot \vec{u} = \|\vec{u}\|^2$, $\vec{v} \cdot \vec{v} = \|\vec{v}\|^2$, and $\vec{u} \cdot \vec{v} = \vec{v} \cdot \vec{u}$.

For Any Vectors u⃗\vec { u }u And v⃗\vec { v }v

For Any Vectors $\vec { u }$ And $\vec { v }$