why vector $mn=|m||n|cos(m,n)$ while vector $ij=0$?

Question

Today I run into some math examples, and I saw that when they calculated vectors $mn$ the formula was ( $mn=|m|\cdot |n|\cdot \cos(m,n)$ ) but there $ij$ were calculated as $ij=0$. So why they don't use the same formula for i and j as well, $ij=|i|\cdot |j|\cdot \cos(i,j)$?

Answer 1

The formula you mentioned gives the dot product of two vectors with an angle $\theta$ between them, $\vec{v}\cdot\vec{u}=|u||v|\cos(\theta)$. The $i$ and $j$ vectors are unit vectors in the $x$ and $y$ directions. Since $x$ and $y$ directions are perpendicular, that is the angle between $i$ and $j$ is 90$^\circ$. Since $\cos(90)=0$, we see that, $i\cdot j=|i||j|\cos(90)=0$. Hope this answers your question.