I am currently taking a course in linear algebra and in my linear algebra book at the chapter about scalar product / dot product it says
"Assume that $u=AB$ and $v=AC$. We now have that $|AC|cos(\theta)=|AD|$ and therefore $u\cdot v=|AB||AC|cos(\theta)=|AB||AD|$".
The figure that illustrates this is a line $AB$ that is visibly longer than a line $AC$. The line $AC$ going above the line $AB$ at an angle of what looks like about 35 degrees. Then there is a dotted line from the end of $AC$ straight down, creating a triangle at point $D$ which is between the points $A$ and $B$.
What is the point of showing that $u\cdot v = u \cdot v_u$? This is clearly $v=v_u$.
In general, when you have $$ x\cdot y=x\cdot z $$ where $x,y,z$ are three vectors (say in the Euclidean space), it does not follow that $y=z$.
This is because $x\cdot w=0$ does not imply "$x=0$ or $w=0$".