# How to determine the vector between projected point and a point along the projection vector?

Problem Setup. As depicted in the image, I have a unit vector $\vec{u}$ between $a$ and $b$ and a point $c$ which I want to project onto the line $a + \lambda \vec{u}$. The projected point onto this line is $d$ which I get from $(\vec{c}\bullet\vec{u})\ \vec{u}$. I wonder how I obtain the vector between $a$ and $d$? I always get strange results, which don't yield the right vector.

Any help is appreciated and thanks in advance!

Edit: Additional Problem Description I guess my calculation of point $d$ is wrong. I always get point $d'$.

Be carefull, $\vec c$ is not the vector you want to project, it's $\vec c - \vec a$. So you have to compute $\vec v=[(\vec c - \vec a)\bullet\vec u] \vec u$
Then to have $\vec d$, just do : $\vec d= \vec a + \vec v$