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?

  • 1
    $\begingroup$ I always found diagrams and pictures useful, so you can see what the operation represents $\endgroup$ – 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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.