Vectors

Vector Representation

• Given $\vec{\mathbf{v}}$ is a vector
  • 2D Plane:
    • Cartesian: $\vec{\mathbf{v}}=(v_x,v_y)$
    • Polar: $\vec{\mathbf{v}}=(v,\theta )$
  • 3D Space:
    • Cartesian: $\vec{\mathbf{v}}=(v_x,v_y,v_z)$
    • Spherical:

2D Plane

Polar $\to$ Cartesian

$\begin{array}{c}\vec{\mathbf{v}}=(v,\theta )\end{array}⟹\{\begin{array}{c}{v}_x=v\cos \theta \\ v_y=v\sin \theta \end{array}$

Cartesian $\to$ Polar

$\begin{array}{c}\vec{\mathbf{v}}=(v_x,v_y)\end{array}⟹\{\begin{array}{c}\Vert \vec{\mathbf{v}}\Vert =v=\sqrt{v_x^2+v_y^2}\\ \theta =\arctan \left(\frac{v_y}{v_x}\right)\end{array}$

3D Space

## Cartesian $\to$ Spherical
$$\begin{array}{c} \vec{\mathbf{v}} = (v_x,v_y,v_z)  \\   \end{array}  \implies \left\{ \begin{array}{c} \| \vec{\mathbf{v}} \|=v = \sqrt{v_x^2+v_y^2+v_z^2} \\ \theta = \arctan\left(\displaystyle \frac{v_y}{v_x}\right) \\ \phi = \arccos\left(\displaystyle \frac{v_z}{v}\right)  \end{array} \right.$$
## Spherical $\to$ Cartesian
$$\begin{array}{c} \vec{\mathbf{v}} = (v,\theta,\phi)  \\   \end{array}  \implies \left\{ \begin{array}{c} v_x = v\sin\phi\cos\theta \\ v_y = v\sin\phi\sin\theta \\ v_z = v\cos\phi  \end{array} \right.$$

Sum of Vectors

• Vector $\mathbf{a}$ has magnitude $a$ and is on the $x$-axis.
• Vector $\mathbf{b}$ has magnitude $b$ and forms an angle $\theta$ with $\mathbf{a}$.
• $\mathbf{a}+\mathbf{b}=(a+b\cos (\theta ),b\sin (\theta ))$
• $\Vert \mathbf{a}+\mathbf{b}\Vert =\sqrt{a^2+2ab\cos (\theta )+b^2}$ (Law of Cosines)

Dot Product (Scalar Product)

Coordinate definition

Given two vectors $\text{a}=(a_1,a_2,...,a_n)$ and $\text{b}=(b_1,b_2,...,b_n)$, the dot product of $\text{a}$ and $\text{b}$ is defined as: $\mathbf{a}\cdot \mathbf{b}=\sum _{i=1}^n{a}_i{b}_i=a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n$

Geometric definition

Given two vectors $\text{a}$ and $\text{b}$, and the angle between them is $\theta$, the dot product is defined as: $\mathbf{a}\cdot \mathbf{b}=\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert \cos \theta$

Matrix Product definition

If $\text{a}$ and $\text{b}$ are identified as column vectors, the dot product can also be written as a matrix product $\mathbf{a}\cdot \mathbf{b}={\mathbf{a}}^T\mathbf{b}$

Properties

• Symmetry $\text{a}\cdot \text{b}=\text{b}\cdot \text{a}$
• Distributive $(\text{a}+\text{b})\cdot \text{c}=\text{a}\cdot \text{c}+\text{b}\cdot \text{c}$
• Homogeneity $(t\text{a})\cdot \text{b}=t(\text{a}\cdot \text{b})$
• Positivity $\text{a}\cdot \text{a}\ge 0$
  • $\text{a}\cdot \text{a}=0⟺\text{a}=\text{0}$
• $\text{0}\cdot \text{a}=\text{a}\cdot \text{0}=0$

Cross Product

• $\mathbf{a}\times \mathbf{b}=\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert \sin (\theta )\mathbf{n}$
  • $\theta$ is the angle between $\text{a}$ and $\text{b}$
  • $\Vert \mathbf{a}\Vert$ and $\Vert \mathbf{b}\Vert$ are the magnitudes of the vectors $\text{a}$ and $\text{b}$
  • $\mathbf{n}$ is a unit vector perpendicular to the plane containing $\text{a}$ and $\text{b}$, with direction s.t. $(\text{a},\text{b},\text{n})$ is positively oriented
• $\mathbf{a}\times \mathbf{b}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|$
• $\mathbf{a}\times \mathbf{b}=-\mathbf{b}\times \mathbf{a}$ (anti-commutative)
• $\mathbf{a}\times \mathbf{a}=0$
• $\mathbf{a}\times (\mathbf{b}+\mathbf{c})=\mathbf{a}\times \mathbf{b}+\mathbf{a}\times \mathbf{c}$
• $\Vert \mathbf{a}\times \mathbf{b}\Vert =\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert \sin (\theta )$, which is the area of the parallelogram spanned by $\text{a}$ and $\text{b}$

Norm of a Vector

• (d12.1.3) The norm (especially the Euclidean norm) of a vector $\text{x}=(x_1,x_2,...,x_n)$ is defined as $\Vert \text{x}\Vert :=\sqrt{{\sum }_{i=1}^n{x}_i^2}=\sqrt{\text{x}\cdot \text{x}}$
• (q12.1.4) $\Vert \text{a}\Vert =0⟺\text{a}=\text{0}$
• Homogeneity (q12.1.5) $\Vert t\text{a}\Vert =|t|\cdot \Vert \text{a}\Vert$
• Cauchy–Schwarz inequality (12.1.4) $|\text{a}\cdot \text{b}|\le \Vert \text{a}\Vert \cdot \Vert \text{b}\Vert$
• (q12.1.7) $|\text{a}\cdot \text{b}|=\Vert \text{a}\Vert \cdot \Vert \text{b}\Vert ⟺\text{a},\text{b}$ are linearly inpendent
• (q12.1.8) Triangle inequality for vectors $\Vert \text{a}+\text{b}\Vert \le \Vert \text{a}\Vert +\Vert \text{b}\Vert$
• Parallelogram Equation for Vectors $\Vert \text{u}+\text{v}{\Vert }^2+\Vert \text{u}-\text{v}{\Vert }^2=2(\Vert \text{u}{\Vert }^2+\Vert \text{v}{\Vert }^2)$

Here, we defined the norm as the Euclidean norm (aka: 2-norm or L2-norm, denoted $\Vert \mathbf{x}{\Vert }_2$) but there are other norms like the 1-norm, $\infty$-norm, etc.

Here, we use the notation $\Vert \mathbf{x}\Vert$, but it is also common to use $|\mathbf{x}|$ for the Euclidean norm

todo other terms like magnitude and length are also used

Orthogonality

• (d12.2.1) orthogonality of two vectors - $\text{a}$ and $\text{b}$ are called orthogonal if $\text{a}\cdot \text{b}=0$ (This relationship is denoted $\text{a}\perp \text{b}$)
• (q12.2.3) Generalized Theorem of Pythagoras: $\text{a}\perp \text{b}⟹\Vert \text{a}+\text{b}{\Vert }^2=\Vert \text{a}{\Vert }^2+{\Vert \text{b}\Vert }^2$
• (d12.2.2) $\mathbf{v}\perp U$ if for all vectors $\mathbf{u}\in U$, $\mathbf{v}\cdot \mathbf{u}=0$

Projection

• The vector projection of $\text{a}$ onto $\text{b}$ is ${\mathrm{proj}}_{\mathbf{b}}(\mathbf{a})=\frac{\mathbf{a}\cdot \mathbf{b}}{\Vert \mathbf{b}{\Vert }^2}\mathbf{b}$ (sometimes denoted ${\text{a}}_{\parallel \text{b}}$)

• The scalar projection of $\text{a}$ onto $\text{b}$ is given by $s=\Vert \text{a}\Vert \cos (\theta )$

• The vector rejection of $\text{a}$ from $\text{b}$ is ${\mathrm{reg}}_{\mathbf{b}}(\mathbf{a})=\mathbf{a}-{\mathrm{proj}}_{\mathbf{b}}(\mathbf{a})$ (sometimes denoted ${\text{a}}_{\perp \text{b}}$)

• The angle between $\text{a}$ and $\text{b}$ is $\theta =\arccos \left(\frac{\text{a}\cdot \text{b}}{\Vert \text{a}\Vert \Vert \text{b}\Vert }\right)$

• The scalar projection of $\text{a}$ onto $\text{b}$ is given by $s=\Vert \text{a}\Vert \cos (\theta )$

• A surface normal (or simply normal) to a surface at point $P$ is a vector $\text{n}$ perpendicular to the tangent plane of the surface at $P$

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TIP

• Vector Space Operations:
  • Scalar Multiplication: ($\text{scalar}\cdot \mathbf{vector}=\mathbf{vector}$)
  • Vector Addition: ($\mathbf{vector}+\mathbf{vector}=\mathbf{vector}$)
• Dot Product: ($\mathbf{vector}\cdot \mathbf{vector}=\text{saclar}$)
• Cross Product: ($\mathbf{vector}\times \mathbf{vector}=\mathbf{vector}$)