:PROPERTIES: :ID: 51f7af57-31ed-4224-ae23-3e8230d908e7 :mtime: 20231010010637 20231007014644 :ctime: 20231007014642 :ROAM_ALIASES: "dot product" :END: #+title: Vector Scalar Product #+filetags: :public:project: * Names The scalar product is also called the dot product. The dot product is one case of an inner product (although the idea of an inner product might be broader). * Definition $\vec{a} \cdot \vec{b} = |a||b| \cos\theta$ * Algebraic Properties of Dot Product - The dot product is commutative, $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a}$. - The dot product distributes over vector addition $\vec{a}\cdot(\vec{b}+\vec{c})=\vec{a}\cdot\vec{b}+\vec{a}\cdot\vec{c}$ - $a \cdot a = |a | ^2$ - $a \cdot b = 0$ if and only if $a$ and $b$ are orthogonal. - $a \cdot b = |a| |b|$ if and only if $a$ and $b$ are parallel. * Abuse of Notation Some authors use an abuse of notation in writing $\vec{a}^{2} =a\cdot a$ instead of $|\vec{a}|^2$. * Dot product of Cartesian vectors - $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$ - $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$ Thus, in any orthonormal coordinates, $a \cdot b = a_{x}b_{x} + a_{y}b_{y} + a_{z}b_{z}$.