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 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: $hat{u} cdot 2hat{v} = 2(hat{u} cdot hat{v})$
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)).
2.
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.
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