:PROPERTIES: :ID: cedb1489-8203-40db-82b7-48031c23f171 :mtime: 20231022073114 20231014010224 20231012011223 :ctime: 20231012011145 :ROAM_ALIASES: "cross product" :END: #+title: vector product #+filetags: :public:project: * Definition The vector product is when you multiply 2 [[id:5864974d-0edf-4757-9b1f-31b159c9aa7a][vectors]] and get a [[id:5864974d-0edf-4757-9b1f-31b159c9aa7a][vector]]. We define $\vec{a} \times \vec{b} = |a| |b| \sin \theta \hat{n}$ where $\hat{n}$ is a normal vector to both $\vec{a}$ and $\vec{b}$. We choose $\hat{n}$ such that $\vec{a}, \vec{b}, \hat{n}$ form a right-handed system. * Right-Handed Coordinates There are two vectors normal to every pair $\vec{a}, \vec{b}$. Which one we choose depends on if our [[id:d02a62fb-1601-48a4-9cf2-bbb2ade26402][coordinate system]] is right-handed or left handed. By convention, we choose our coordinates to be right handed. If $\vec{a}$ points along your index finger, and $\vec{b}$ points along your middle finger, then $\vec{a} \times \vec{b}$ will point along your thumb (in a right-handed coordinate system). * Relation with the dot product The vector product is often contrasted against the [[id:51f7af57-31ed-4224-ae23-3e8230d908e7][dot product]], because the [[id:51f7af57-31ed-4224-ae23-3e8230d908e7][dot product]] returns a scalar and the vector product returns a [[id:5864974d-0edf-4757-9b1f-31b159c9aa7a][Vectors]]. * Other Names for Vector Product The vector product is also called the "cross product" or the "wedge product". * Algebraic Properties of Vector Product - Anti-commutative: $\vec{a} \times \vec{b} = - \vec{b} \times \vec{a}$ - $|\vec{a} \times\vec{b} = |\vec{b} \times \vec{a}|$ - Distributes over addition: $\vec{a} \times (\vec{b} + \vec{c}) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}$ - NOT associative: $\vec{a} \times (\vec{b} \times \vec{c}) \neq (\vec{a} \times \vec{b}) \times \vec{c}$ - cancellation: $\vec{a} \times \vec{b} = \vec{0}$ if and only if $\vec{a}$ and $\vec{b}$ are parallel or antiparallel. - Idempotence: $\vec{a} \times \vec{a} = \vec{0}$. - Orthogonal: If $\vec{a}$ and $\vec{b}$ are orthogonal, then $|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}|$ * Calculating Vector Product using Determinant It is possible to calculate the vector product of 2 vectors using the [[id:3aadb278-8b51-4ce7-9ef2-8a27fae2349f][determinant]] of a matrix. The way you do this is to calculate $\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a & b & c \\ x & y & z \end{vmatrix}$