## Operations
### Projection
$$proj_{u}v=\frac{u\cdot v}{\left|\left|u\right|\right|^{2}}u$$
Used to project vector $v$ onto $u$
![[VectorProjection.excalidraw]]
### Unit Vector
$$ unit_u=\frac{u}{||u||}$$
Used to find the vector of magnitude 1 in the same direction as $u$
### Cross Product
$$a \times b = \begin{vmatrix} i&j&k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}= i(a_2b_3-a_3b_2)-j(a_1b_3-a_3b_1)+k(a_1b_2-a_2b_1)$$
Used to create a vector that is orthogonal to both $a$ and $b$. To find what direction it will point in use the right-hand rule (thumb, pointer and middle finger).

Properties:
- Not commutative $$u\times v \ne v \times u$$
- Anti-commutative $$ u \times v = -(v \times u) $$
- On self $$ a \times a = 0 $$
The magnitude of the cross product of two vectors can be found with $$||u \times v|| = ||u|| * ||v|| * \sin(\theta)$$
The magnitude of the cross product is equal to the