Vector

An element of a Vector Space, represented in that space either by a direction and magnitude, or by a list of Numbers, each describing the placement along a coordinate axis.

Vector Space Element

Note that Vectors need not be $n\times 1$. Any possible element of a vector space is by definition, a vector. This means that vector spaces really contain Tensors. 99% of the time when people are talking about vectors, they really mean “Coordinate Vector”, i.e. a vector that represents a location in Space. Whereas a vector represents a location in a Vector Space.

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Operations

Addition

Add elementwise.

Subtraction

Subtract elementwise.

Scalar Multiplication

Multiply elementwise.

Magnitude

The magntiude calculates the distance from the tip of the vector to the origin.

$$||\vec{u}||=\sqrt{\vec{u}\cdot \vec{u}}=\sqrt{u_1^2+\cdots +u_n^2}$$

Dot Product

Dot Product

Measures the alignment of its operand vectors on a scale of 0 to 1

• $\vec{u}\cdot \vec{v}={\vec{u}}^T\vec{v}$
• $\vec{u}\cdot \vec{v}=\vec{v}\cdot \vec{u}$
• $(\vec{u}+\vec{v})\cdot \vec{w}=\vec{u}\cdot \vec{w}+\vec{v}\cdot \vec{w}$
• $(c\vec{u})\cdot \vec{v}=c(\vec{u}\cdot \vec{w})$
• $||c\vec{v}||=|c|||\vec{v}||$
• $\vec{u}\cdot \vec{u}\ge 0$
• $\vec{u}\cdot \vec{u}=||\vec{u}|{|}^2$
• $\vec{a}\cdot \vec{b}=|a||b|cos\theta$

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Cross Product

Cross Product

$$\vec{a}\times \vec{b}=\left(|a||b|\sin (\theta )\right)\vec{n}=\vec{c}$$

where $\vec{n}$ is the Unit Normal Vector of the Plane described by $\vec{a}$ and $\vec{b}$

Properties

$$\begin{array}{rl}\vec{a}\times \vec{b}& =-(\vec{b}\times \vec{a})\\ \vec{a}\times (\vec{b}+\vec{c})& =\vec{a}\times \vec{b}+\vec{a}\times \vec{c}\\ (\vec{a}+\vec{b})\times \vec{c}& =\vec{a}\times \vec{c}+\vec{b}\times \vec{c}\\ (C\vec{a})\times \vec{b}=\vec{a}\times (C\vec{b})& =C(\vec{a}\times \vec{b})\\ \vec{a}\times (\vec{b}\times \vec{c})& =\vec{b}\times (\vec{c}\times \vec{a})=\vec{c}\times (\vec{a}\times \vec{b})\end{array}$$

Magnitude

$$\underset{\text{Area}({▰}_{a,b})}{\underbrace{|\vec{a}\times \vec{b}|=|a||b|\sin (\theta )}}⟹\vec{a}\parallel \vec{b}⟺\vec{a}\times \vec{b}=\vec{0}$$Link to original