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My textbook is in Swedish so I am sorry if the wording is a bit weird or if things have got another name in english. Anyway, I am confused regarding Scalar Multiplication and Orthogonal projection, especially how scalar multiplication is used to derive the formula for orthogonal projection.

Scalar multiplication is defined as: $$|u| \cdot |v| \cdot \cos [u,v]$$ Orthogonal projection is defined as $$u' = \frac{u\cdot v}{|v|^2} \cdot v$$

From what I understand the difference between these is that scalar multiplication produces a magnitude, while orthogonal projection produces a vector.

I am trying to prove the formula for orthogonal projection myself and I have come up with the follwing:

Define $$ e = \frac{1}{|v|} \cdot v$$ i.e a normalized vector in the direction of v. Now, the magnitude times the direction would produce the full vector correct? So:

$$ u' = |u|\cdot|v|\cdot\cos [u,v] \cdot e = \frac{|u|\cdot|v|\cdot\cos [u,v] \cdot v}{|v|}$$

Which gives $$\frac{u\cdot v\cdot v}{|v|}$$

Which is not the formula for orthogonal projection. Where am I going wrong?

Michael Hardy
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user262493
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  • You should distinguish between the different types of multiplication you have going on. For instance in the equation $u' = \dfrac{uv}{|v|^2} v$, the first $$ should be the dot product/ scalar product, whereas the second $$ should denote scaling the vector $v$ by the scalar quantity in front of it. So maybe a better way to write it would be $$u' = \frac{u \cdot v}{|v|^2} v$$ –  Dec 02 '15 at 20:26
  • @Bye_World Yeah sorry I am quite new to linear-algebra so I am not familiar with the notation. – user262493 Dec 02 '15 at 20:28
  • Here and here are two resources that might help you see where the orthogonal projection formula come from. –  Dec 02 '15 at 20:32
  • Your problem is in thinking that the length of $u'$ should be $|u||v|\cos(\theta)$. It should actually just be $|u|\cos(\theta)$ (try to draw a picture to confirm this). So then you end up with $u'= |u|\cos(\theta)e = \frac{|u|\cos(\theta)}{|v|}v$. Then if you multiply this by $1=\frac{|v|}{|v|}$ you get the orthogonal projection formula. –  Dec 02 '15 at 20:43

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