• A vector is an object that has both a magnitude and a direction.
• It’s a line segment, whose length (Actual Drawing Length) is the magnitude of the vector and with an arrow indicating the direction.

• Two vectors are considered the same if they possess both the same magnitude (length) and direction (parallel).
• The specific location or starting point of a vector does not affect its essential properties.
• Two examples of vectors are force and velocity. Both are in a particular direction. The magnitude of the vector indicates the strength of the force or the speed associated with the velocity.
• The number of components of the vector depends on the dimensionality of the space. More than 3D cannot be imagined by human brains.

Vector Notation

• We denote vectors using boldface as in a or b
• But, if handwriting vectors, people denote them as
  • $\vec{a}$ or $\vec{b}$
  • $\hat{a}$ or $\hat{b}$
  • $\overline{a}$ or $\overline{b}$
• We denote the magnitude of the vector a by $∥a∥$.

Different Vector Representations

There’re several different ways to represent vectors, depending on the context and the mathematical framework being used

1. Geometric Representation
   • This means that we will represent it by graphically drawing it.
   • We discussed it above, using line segments for length and arrows for directions.
2. Vector Representation
   • In a 2D cartesian coordinate system, the vector can be represented as (x, y). For example: $v = (3, -2)$.
   • In a 3D cartesian coordinate system, the vector can be represented as (x, y, z). For example: $v = (3, -2, -1)$.
   • We start at any point in the Euclidean space, then we move 3 steps to the +ve x-axis then two steps down to the -ve y-axis, then one step down to the -ve z-axis.
     • The point we stop at is the final point
     • Now connect from the first point to the final point, this is what we meant by $v = (3, -2)$.
   • Parentheses: Using ( ) is a widely accepted convention in Vector Notation —— $v = (3, -2)$.
   • Angle Brackets: Using < > is also commonly used to represent vectors, especially in mathematical literature and textbooks —— $v = \langle3, -2\rangle$
   • It’s often better to represent vectors using their coordinations as it’s easier to manipulate rather than manipulating its magnitude and direction directly.
   • The number of components of the vector depends on the dimensionality of the space. More than 3D cannot be imagined by human brains.

1. Matrix Representation
   • Vectors can also be represented as column matrices or row matrices.
   • For example, a vector $v = (3, 4)$ in 2D space can be represented as a column matrix:
     $$\begin{bmatrix} 3 \\ 4 \\ \end{bmatrix}$$
   • And can be represented as a row matrix:
     $$\begin{bmatrix}3 & 4 \\\end{bmatrix} \text{ OR }\begin{bmatrix}3 ; 4 \\\end{bmatrix}$$
2. Polar Representation (Magnitude & Direction)
   • In polar coordinates , a point is described by its radial distance from the origin (magnitude) and the angle it forms with a reference direction (direction).
   • This is similar to how vectors can be represented using their magnitude and direction —> So we call it polar representation.
   • A vector $v$ with magnitude $∥v∥$ and direction $θ$ can be expressed as $∥v∥∠θ$
   • How to calculate the Vector Magnitude?
3. Component Representation

Vector Magnitude

• Vector Length == Vector Magnitude == $|v|$
• It’s a non-negative scalar value.
• It represents the distance from the origin to the tip of the vector when it is represented graphically.
• IMPORTANT: To calculate the vector magnitude we use a distance formula!

The normal Distance Approach (L2 Norm) (Euclidean Distance)

In Euclidean n-space, the distance between two points is calculated using the Euclidean distance formula, which is a generalization of the Pythagorean theorem. The Euclidean distance formula accounts for the additional dimensions by summing the squares of the differences along each dimension and taking the square root of the result.

Vector Magnitude

1. Euclidean Norm: For vectors represented as lists of components, such as (x, y, z) in 3D space, the Euclidean norm or 2-norm is often used. It is calculated using the square root of the sum of the squares of the vector's components:

   |v| = √(v₁² + v₂² + v₃² + ... + vn²)

2. Vector Dot Product: The magnitude of a vector can be calculated using the dot product operation. Given a vector v, the dot product v ⋅ v (v dot v) yields the square of the vector's length. Therefore, the magnitude can be obtained as:

   |v| = √(v ⋅ v)

3. Others are Manhattan or absolute value??
4. Coordinates: Points are typically represented by coordinates in a coordinate system, indicating their position in space. Vectors also have components in a coordinate system, but these components represent the direction and magnitude of the vector rather than a specific location.

• points and vectors are both just lists of numbers, but vectors are coordinate-free.
• Why make the distinction between points and vectors?
  • A vector need not start at the origin: it can be located anywhere! In other words, an arrow is determined by its length and its direction, not by its location.
• For example, all these arrows represent the vector $\left\langle 1, 2 \right\rangle$

  

• Another way to think about a vector is as a difference between two points, or the arrow from one point to another. For instance, $\left\langle 1, 2 \right\rangle$ is the arrow from (1,1) to (2,3)

  

The point with coordinates (0,0,…,0) is called the origin of coordinates. It is customary to adopt it as the common point of origin for all vectors in an N-dimensional space, although it is not mandatory.

1. Polar Representation: In polar coordinates, vectors can be represented using their magnitude and angle. The magnitude represents the distance from the origin, and the angle represents the direction or orientation of the vector with respect to a reference axis.
2. Parametric Representation: Vectors can be represented using parametric equations. In this representation, each component of the vector is expressed as a function of a parameter. This allows for a flexible way to describe vector quantities in terms of other variables.
3. Basis Vector Representation: In linear algebra, vectors can be represented as linear combinations of basis vectors. A basis is a set of linearly independent vectors that span a vector space. Vectors are then expressed as a sum of scalar multiples of the basis vectors.

1. Linear Algebra Definition: In linear algebra, vectors are elements of a vector space. A vector space is a mathematical structure that satisfies certain properties, including closure under vector addition and scalar multiplication. Vectors in linear algebra can be represented as columns or rows of numbers and are subject to various operations, such as vector addition, dot product, and cross product.

Similarities between points and vectors

• Point identifies a specific location — vector indicates a direction from one point to another.
• Coordinates: Points and vectors can be described using coordinate systems. In 2D space, they can be represented by ordered pairs (x, y), where x and y are the coordinates of the point or the components of the vector.

1. Translation: Points and vectors can be translated or moved within a coordinate system. Translating a point involves shifting it by a vector, which indicates the displacement. The result is a new point with modified coordinates. Similarly, applying a translation vector to a vector does not change its direction but results in a new vector pointing to a different location.
2. Geometric Operations: Both points and vectors can be subject to geometric operations. For example, vector addition allows combining vectors to obtain a resultant vector, while point subtraction can yield a vector representing the displacement between two points.
3. Magnitude: Although points don't have a magnitude, vectors do. The magnitude of a vector represents its length or size, indicating the distance between its initial and terminal points. This concept is not applicable to points, as they do not possess magnitude.

While points and vectors have similarities, it's important to note their fundamental differences: points represent specific locations in space, while vectors represent direction and magnitude. Points are fixed, whereas vectors can be moved or applied to different locations. Understanding these distinctions is crucial for utilizing points and vectors effectively in mathematical and geometric contexts.

Similarities:

1. Geometric Representation: Both points and vectors can be represented graphically using symbols, such as dots for points and arrows for vectors, to visualize their positions and directions.
2. Line Segments: Line segments can be formed by connecting two points or by using a vector as a displacement between two points. In both cases, they represent a straight path between the two locations.
3. Scaling: Both points and vectors can be scaled or multiplied by a scalar value, resulting in a change in magnitude. This operation stretches or shrinks the vector or displaces the point from the origin.

Differences:

1. Conceptual Meaning: Points represent specific locations or positions in space, whereas vectors represent direction and magnitude. Points have no inherent notion of displacement or direction.
2. Operations: Points and vectors are subject to different mathematical operations. Points can be added or subtracted, but these operations result in new points. Vectors, on the other hand, can undergo operations like addition, subtraction, scalar multiplication, and dot product, which yield new vectors.
3. Transformation: Points are subject to transformations such as translation, rotation, and scaling, which change their position and orientation. Vectors, however, are not affected by translation or position changes since they are based on direction and magnitude.
4. Magnitude vs. Location: Vectors have a magnitude that represents their length, while points have no inherent magnitude. The distance between two points can be calculated, but it is not a property of the points themselves.

Unit Vectors

Yes, unit vectors do have directions. A unit vector is a vector with a magnitude of 1, and it indicates the direction of a particular vector quantity. Unit vectors are often used to describe the orientation or direction of other vectors in mathematics and physics. By definition, a unit vector is dimensionless and retains only information about direction.

Unit vectors do have direction. A unit vector is a vector with a magnitude of 1, but it still retains information about direction. Unit vectors are commonly used in coordinate systems to represent and manipulate vectors. They serve as a basis for describing the direction of other vectors by specifying their components along different coordinate axes. By normalizing a vector (dividing it by its magnitude), we obtain a unit vector that points in the same direction but has a magnitude of 1. Unit vectors are crucial for performing vector operations such as addition, subtraction, and dot product calculations.

Vector Space

• Also called “Linear Space”