ObsidianVault/Wiki/Math/Vector Basics.md

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