Question

Show that if the vectors v, w are orthogonal then ||v+w||^2 = ||v||^2 + ||w||^2

Answer 1

\[\Large \vec{v}\cdot \vec{v}=vv\cos0=v^2 \]
so
\[\Large(\vec{v}+\vec{w})\cdot(\vec{v}+\vec{w})=|| \vec{v}+\vec{w}||=\vec{v}\cdot\vec{v}+2\vec{v}\cdot \vec{w}+\vec{w}\cdot\vec{w} \]

Answer 2

Know how to complete from here?

Answer 3

No I'm completely lost

Answer 4

Well I only applied the definition of length through the scalar product and expanded it from one vector to two vectors (which happen to be orthogonal to one another), orthogonal means that:
\[\Large \vec{v}\cdot \vec{w}=0 \]
As by the scalar product:
\[\Large \vec{a}\cdot \vec{b}=||\vec{a}|| \cdot ||\vec{b}|| \cos\gamma \]

for gamma=90 degree, the right hand side becomes zero, therefore orthogonal vectors must turn out to be zero when scalar multiplied.