# How to teach relation between scalar product and line projection

The scalar product $\vec a \cdot \vec b$ of two vectors $\vec a, \vec b \in \mathbb R^n$ equals the length of the orthogonal projection of $\vec a$ to the linear span of $\vec b$, and vice versa.

How to teach this relation to students that only know the very basics of finite-dimensional vector spaces?

• I always found diagrams and pictures useful, so you can see what the operation represents – John Doe May 28 '17 at 10:57

You could teach the relationship between the scalar product and the modulus. You have that, if $\mathbf{a}$ and $\mathbf{b}$ are collinear their scalar product is the product of the moduli, and if $\mathbf{a}$ and $\mathbf{b}$ are perpendicular their scalar product is 0 (you can show this through exemples), then you should have $\mathbf{a} \cdot \mathbf{b}=||\mathbf{a}|| \cdot ||\mathbf{b}||\cdot f(\theta)$ where $f(\theta)$ is some function of the angle between the two vectors, which is between 0 and 1 so that $f(0)=1$ and $f\left(\frac{\pi}{2}\right)=0$, intuitively, they can imagine that this function is $\cos(\theta)$, and then you can use a basic trigonometry argument for the projection.