Finding the Orthogonal Projection of a Vector onto Another Vector

[linuspauling]

Given:

\vec A \cdot \vec B = non zero
and
\theta does not equal 0

I can't seem to prove that Vector B minus the Projection of B onto A makes the orthogonal projection of B onto A.

Can you help?

 

[arunbg]

I am not sure I understand your question properly. The first statement in fact implies the second statement in your question.
You can see the proof by simple visualization, using laws of vector geometry of course.
Alternatively, prove that the dot product of "Vector B - Projection of B onto A and vector A" is 0. Just put that statement in algebra and simplify.

 

[HallsofIvy]

The projection of B onto A is
\frac{\vec{A}\cdot\vec{B}}{|\vec{A}|^2}\vec{A}
so
(\vec{B}- proj_{\vec{B}}(\vec{B}))\cdot \vec{A}= \vec{A}\cdot\vect{B}- \frac{\vec{A}\cdot\vec{B}}{|\vec{A}|^2}\vec{A}\cdot\vec{A}

 

[linuspauling]

thank you halls of ivy.

thanks hall of ivy