I am currently doing a research project into quaternion. I have looked at the proof of quaternion rotation identity as outlined by Wikipedia on https://en.wikipedia.org/wiki/Quaternions_and_spatial_rotation.
The part that I am confused about is the last line of the proof. How is v(perpindicular)= v-u(u \cdot v)? I understand how the other expression is the parallel component.
Many thanks
This is just the standard formula for resolving a vector into its components along and perpendicular to a given (nonzero) vector. The "along" component is usually called the projection. The other bit is the orthogonal component. The projection of $v$ along $u$ is given by the well-known formula $$ \mathrm{proj}_u(v) = \frac{u \cdot v}{u \cdot u}u. $$ Drawing out the triangle rule for vector subtraction, you get that the perpendicular component is just $$ v_\bot = v - \mathrm{proj}_u(v) = v - \frac{u \cdot v}{u \cdot u}u. $$ But your $u$ is a unit vector, so $u \cdot u =1$, so this becomes $$ v_\bot = v - (u \cdot v)u $$ which is what you wanted.