dot product

Dot product

$$a\cdot b=||a||||b||cos(\theta )$$

$\theta$ … angle between $a$ and $b$
equivalent to:

$$a\cdot b=a_1{b}_1+\text{... }+a_n{b}_n$$

https://youtu.be/tdwFdzVqito (cosine formula, dot vs cross) 3b1b!!!

“Project” one vector onto another and multiply the length of the other with the resulting vector.
The head of the projected vector follows from drawing an orthogonal line from the head of one vector to another.
Order does not matter.

Dot product is positive if they point into the same direction, negative for opposing directions:

The projected vector $\vec{w}$ has length $0$ if the vectors are perpendicular:

Indifference of order:

Scaling one of those vectors by 2 → dot product is exactly 2x bigger:

$$(2\vec{v})\cdot \vec{w}=2(\vec{v}\cdot \vec{w})$$

Intuition for the cosine formula
khan ac yt

Multiplying the length of $\vec{b}$ times the amount of $\vec{a}$ thats going in the same direction as $\vec{b}$

If they are perfectly parallel, the dot-product is simply $||\vec{a}||||\vec{b}||\cos (90)$, so simply their lengths multiplied!

TODO go through this rq on paper

See also Matrix Vector Product Intuition