Vector orthogonal projection what's my error

I've been recalculating the following problem for hours and I don't understand what I'm doing wrong, please help me.

We have the following figure:

By using orthogonal projection, determine the length of segment AD. (Segment BD and AC are perpendicular)

Here's what I tried:

\begin{align*} AB &= (0,10,6) - (-2,8,6) = (2,2,1)\\\\ AC &= (4,5,5) - (-2,8,5) = (6,-3,0)\\\\ CB &= (0,10,6) - (4,5,5) = (-4,5,1)\\\\ CD &= \frac{CB \cdot AC}{\|AC\|^2} \cdot AC = \frac{(-24 + -15) - 39 }{\sqrt{45}} \cdot (6,-3,0)\\\\\\ &\text{(Does not make any sense from here)}\\\\\\ CD &= (-34.88, 17.44, 0)\\\\ D &= (-30.88, 22.44, 5)\\\\ AD &= \sqrt{-30.88} \end{align*}

How can I solve this problem?

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1 Answer

To project $CB$ in the direction of $CA$, you could use: $$\|CD\| = \frac{CB \cdot CA}{\|CA\|}$$

and using $\|AD\| = \|AC\| - \|CD\|$, you should get $\dfrac{2}{ \sqrt{5}}$ after a bit of simplifying.

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Alright, it gave me this answer once but I didn't trust myself since this problem is divided in 4 question. And on the next question they are saying that the length of CE = 2, this does not make much sense if we look at the picture – Machinegon Mar 29 '13 at 18:57