Related to dot product and vectors

Question

For any vector $r$, prove that $$r = (r.i)i + (r.j)j + (r.k)k$$ I cannot satisfied myself with the solution in picture-https://photos.app.goo.gl/hjdqt93dSEq69cSX8 Since, $$(r.i)i=r(i.i)=r|i|²=r$$ Similarly the other two terms gives $r$. So, $$(r.i)i + (r.j)j + (r.k)k=3r$$ This is not equal to $r$. So what is wrong in this solution?

Answer 1

The error is in the step

$$ (r\cdot i)i = r(i\cdot i)$$ This just isn't correct. For instance, suppose $r=j$. Then $j\cdot i = 0$, $$0= (j\cdot i)i \overset{\Huge!?}=j(i\cdot i) = j|i|^2 = j$$

The dot product doesn't work like this. Remember that $r\cdot i$ is a number. Any number $s$ times the vector $i$ gives $si$, a vector pointing in the direction $i$. On the other hand, $(i\cdot i)r$ is some other number $l$ times $r$, resulting in $lr$, a vector pointing in the direction of $r$. The numbers $r\cdot i,r\cdot j, r\cdot k$ are just the components.

Example- if $r = \begin{pmatrix}3\\1\\2\end{pmatrix}$, then $$r\cdot i=3,r\cdot j=1, r\cdot k=2$$