# Why is the dot product of perpendicular vectors zero?

I've read that taking a dot product is just projecting one vector on the other, so a perpendicular vector will have no components in the other vectors direction. But shouldn't this leave the length unchanged so it has its original magnitude like multiplying it by 1?

• "taking a dot product is just projecting one vector on the other" What? – Angina Seng Jan 4 at 20:50
• When does the cosine rule reduce to the Pythagorean theorem? – Karl Jan 4 at 21:33

$$u.v=|u||v|\cos \theta$$
If $$\theta =\pi/2$$ we have $$\cos \theta =0$$
To clarify, the projection of $$\vec u$$ on $$\vec v$$ is the vector $$\left(\frac{\vec u\cdot \vec v}{\|\vec v\|^2}\right)\vec v = \left(\frac{\vec u\cdot\vec v}{\|\vec v\|}\right)\frac{\vec v}{\|\vec v\|}.$$ The dot product is a scalar quantity. But the length of the projection is always strictly less than the original length unless $$\vec u$$ is a scalar multiple of $$\vec v$$.