Vectors

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: ($\mathbf{vector}\times \mathbf{vector}=\mathbf{vector}$)

Dot Product

also scalar product or Euclidean inner product

Definition:

• Coordinate definition: $\mathbf{a}\cdot \mathbf{b}=\sum _{i=1}^n{a}_i{b}_i=a_1{b}_1+a_2{b}_2+\cdots +a_n{b}_n$
• Geometric definition: $\mathbf{a}\cdot \mathbf{b}=\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert \cos \theta$
• Matrix definition: $\mathbf{a}\cdot \mathbf{b}={\mathbf{a}}^T\mathbf{b}$
• Projection definition: $\mathbf{a}\cdot \mathbf{b}=\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert \cos \theta$
  • Where the projection of $\text{a}$ onto $\text{b}$ is ${\mathrm{proj}}_{\mathbf{b}}(\mathbf{a})=\frac{\mathbf{a}\cdot \mathbf{b}}{\Vert \mathbf{b}{\Vert }^2}\mathbf{b}$
  • And the length of $\text{b}$ is $\Vert \mathbf{b}\Vert$
  • So, $\mathbf{a}\cdot \mathbf{b}=\Vert \mathbf{a}\Vert \Vert \mathbf{b}\Vert \cos \theta$

Properties: (12.1.2)

• 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 $\mathbf{u}\in U$, $\mathbf{v}\cdot \mathbf{u}=0$