Vector

Description

Starting point doesn't mean for Vectors
Vectors will show by lowercase letters and an arrow on top of it or just with a bold lowercase letter
Vector = Magnitude + Direction
Velocity = Magnitude + Direction
So indeed Velocity is a Vector
Standard Position (Origin): \((0, 0)\) is the standard position so when one the tail of one Vector is in this position we will say that this Vector is in the standard position

N Dimensional Real Coordinate Space

Tuple: means a list of numbers.
Cartesian coordinate system: is the specific name for 2-tuples.
3-tuples: means 3 numbers in one matrix.

Operations

Addition

All the purple arrows are the same
All the green arrows are the same
The starting point doesn't matter in Vectors
The blue arrow completely shows us how adding two Vectors to each other works in the background

Multiplication

Subtraction

Dot Product

Will tell us how much our vectors are moving together.

\(\vec{w} \cdot \vec{x}\) = \(w_1x_1 + w_2x_2 + w_3x_3 + \dots + w_nx_n\)

The order doesn't matter in a dot product: \(\vec{v} \cdot \vec{w} = \vec{w} \cdot \vec{v}\)
Distribution is applicable on the dot product:

\((\vec{v} + \vec{w}) \cdot \vec{x} = (\vec{v} \cdot \vec{x} + \vec{w} \cdot \vec{x})\)

\((c \vec{v}) \cdot \vec{w} = c (\vec{v} \cdot \vec{w})\)

Cross Product

Will give us the normal vector of the space (plane) of two other vectors (\(\vec{a}\) and \(\vec{b}\)).

\[ \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \times \begin{bmatrix} b_1 \\ b_2 \\ b_3 \end{bmatrix} = \begin{bmatrix} a_2 b_3 - a_3 b_2 \\ a_3 b_1 - a_1 b_3 \\ a_1 b_2 - a_2 b_1 \end{bmatrix} \]

The cross-product is only applicable in \(R_3\).

Length (Magnitude)

Let the vector \(\vec{a}\) be represented as:

\[ \vec{a} = \begin{bmatrix} a_1 \\ a_2 \\ \vdots \\ a_n \end{bmatrix} \]

The magnitude or length of vector \(\vec{a}\) can be calculated as:

\[ \|\vec{a}\| = \sqrt{a_1^2 + a_2^2 + \dots + a_n^2} \]

Alternatively, using the dot product, the magnitude can be expressed as:

\[ \|\vec{a}\| = \sqrt{\vec{a} \cdot \vec{a}} \]

Unit Vector

Unit Vector of Axes

Unit Vector of a Vector

To find a unit vector, u, in the same direction of a vector, v, we divide the vector by its magnitude.

\[ \vec{u} = \frac{\vec{v}}{\|\vec{v}\|} = \frac{1}{\|\vec{v}\|} \vec{v} \]

For a vector \(\vec{v} = \begin{bmatrix} a & b \end{bmatrix}\) its magnitude is given by

\[ \|\vec{v}\| = \sqrt{a^2 + b^2} \]

Cauchy–Schwarz Inequality

\[ |\vec{x} \cdot \vec{y}| \leq \|\vec{x}\| \, \|\vec{y}\| \]

\[ |\vec{x} \cdot \vec{y}| = \|\vec{x}\| \, \|\vec{y}\| \iff \vec{x} = c\vec{y} \]

Triangle inequality

\[ \|\vec{x} + \vec{y}\| \leq \|\vec{x}\| + \|\vec{y}\| \]

Perpendicular Vectors (Orthogonal Vectors)

The vectors \(\vec{x}\) and \(\vec{y}\) are perpendicular if, and only if, their dot product is equal to zero: \(\vec{x} . \vec{y} = 0\)

Normal Vector

A normal is an object such as a line, ray, or vector that is perpendicular to a given object.

Eigenvectors & Eigenvalues

Is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted by lambda, is the factor by which the eigenvector is scaled.

Matrix \(A\) acts by stretching the vector \(\vec{x}\), not changing its direction, so \(\vec{x}\) is an eigenvector of \(A\).