Vectors

Vectors
Vectors are lists of points that represent a linear combination of basis vectors... but that's a topic for linear algebra. In two dimensions, they're "arrows" from the origin to a point. So, the vector $\langle 3, 2\rangle$ represents "3 units across and two up."

There are some nontrivial operations we can do with vectors (I'm sure you can figure out how to do vector addition and multiplication by scalars).

Dot Product ($\vec{a}\cdot \vec{b}=c$)
This is the "multiplication" of vectors. Essentially, it gives us a scalar value representing how much $\vec{a}$ aligns with $\vec{b}$ in terms of magnitude and direction.

There are two ways to calculate it: $\vec{a}\cdot\vec{b}=||a|| \cos(\theta)$ and $\langle a_x a_y \rangle\cdot \langle b_{x},b_{y}\rangle=a_{x}b_{x}+a_{y}b_{y}.$

Projection
The projection of a vector $\vec{a}$ onto another vector $\vec{b}$ gives us the actual vector: $$\text{proj}_{b}(\vec{a})=\frac{a\cdot b}{||b||^2} \vec{b}.$$ See if you can figure out why this makes sense given what you know about the dot product (and the diagram above).

Cross Product ($\vec{a}\times \vec{b}=\vec{c}$)
In three dimensions, the cross product of $\vec{a}$ and $\vec{b}$ returns a vector $\vec{c}$ that is perpendicular to both — that is, the normal vector to the plane defined by $\vec{a}$ and $\vec{b}.$ The magnitude of $\vec{c}$ is the area of the parallelogram with $\vec{a}$ and $\vec{b}$ for sides. Calculating it is somewhat tedious, but it looks like: $$\vec{a}\times \vec{b}=(a_{1} \hat{i}+a_{2} \hat{j} + a_{3} \hat{k})\times (b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k})= \begin{bmatrix}a_{2}b_{3}-a_{3}b_{2}\\ a_{3}b_{1} - a_{1}b_{3} \\ a_{1}b_{2} -a_{2}b_{1}\end{bmatrix}.$$ So... I hope and pray that you don't have to calculate this manually in any of your classes, ever.