Reference: “Dot products and duality | Essence of linear algebra, chapter 9”, 3Blue1Brown

Algebra example:

$\begin{bmatrix} 1 & 2 \end{bmatrix} \cdot \begin{bmatrix} 3 \ 4 \end{bmatrix} = 1 \cdot 3 + 2 \cdot 4$

Gemoetrically:


The dot product of v and u = (length of projection of v onto u)(length of u)

Order and Symmetry

Order doesn’t matter. (Length of projected v) x (Length of u) = (Length of projected u) x (Length of v)

Why? Symmetry

Symmetry Example: Suppose we have anoher two unit vectors $\hat{v}$ and $\hat{u}$ . Then, the length of projection of $\hat{v}$ onto $\hat{u}$ = the length of project of $\hat{u}$ onto $\hat{v}$. This is the symmetry between the two vectors.


Now we scale $\hat{v}$ by 2, we have 2$\hat{v}$. The projection of $\hat{u}$ on $\hat{v}$ does not change, while the projection of $\hat{v}$ is doubled. Hence the symmtery is broken. However, the dot product is simply scaled by 2, too. Before the scale, the dot product of v and u is: $\hat{u} \cdot \hat{v}$ After the scale, the dot product of v and u is: $\hat{u} \cdot 2\hat{v} = 2(\hat{u} \cdot \hat{v})$


Therefore, (v times w) or (w times v) does not matter.

Why dot product is projection?

Suppose I have a number line and a system S that projects vectors(x, y) on the number line:

$S = \begin{bmatrix} ? & ? \end{bmatrix}$

$\begin{bmatrix} ? & ? \end{bmatrix} \cdot \begin{bmatrix} x \ y \end{bmatrix}$

Suppose there are two unit vector $\hat{u}$ and it lies on a number line. $L(v)$ is the linear transformation that maps vector v (x, y) between 2D (a 2D space) and 1D (a number line)).

Suppose we have another two unit vectors $\hat{a}$ and $\hat{b}$, and we use $L$ to map them on the number line. Then the projection between $\hat{a}$ on $\hat{u}$, $\hat{b}$ on $\hat{u}$ are the same, because of unit vectors and the symmetry.

And, the projection happens to be equal to the dot product.