Vectors and Vector Spaces

Introduction to Vectors

• Vectors are used in physics to represent displacement, velocity, momentum, etc.
• Scalars are used to represent distance, speed, mass, etc.

Vector Spaces

• Vectors can exist in 2-space, 3-space, and n-space.
• R represents the set of real numbers.
• R¹ is the 1-dimensional real line.
• R² is the 2-dimensional xy-plane (2-space).
• R³ is the 3-dimensional xyz coordinate system (3-space).
• Rn denotes n-space, where points are given by n coordinates: (x1, x2, x3,... , xn).

Ways to Conceptualize Vectors

• Vectors can be visualized as arrows in n-space
• Vectors can be visualized as points in n-space
• Vectors can be represented as n-tuples of numbers
• The standard basis vectors are unit vectors with length 1
• In R², i = (1, 0) and j = (0, 1)
• Any vector in R² can be written as a linear combination of i and j
• In R³, i = (1, 0, 0), j = (0, 1, 0) and k = (0, 0, 1)

Vector Definition

• A vector in Rn is an ordered n-tuple of real numbers, x = (x1, x2,..., xn).
• Vectors can be written in row form: x = (x1, x2,... , xn)
• Vectors can be written in column form: x =

[x1]
[x2]
[xn]

• Vectors are represented in bold type to distinguish from scalars
• When writing by hand, use an underline or arrow

Vector Sketching

• To sketch a vector x = (a, b) in 2-space, start at an initial point
• Find the second point by moving 'a' units right (left if negative) and 'b' units up (down if negative)
• Connect the two points with an arrow from initial to terminal point
• Sliding vectors to different initial points doesn't change the vector

Vector Addition

• Two vectors can be added together to create a new vector
• Algebraically, x + y = (x1 + y1, x2 + y2,... , xn + yn) in n-space
• Geometrically, vectors are added "head to tail."
• Example: x = (1, 2, 3, 4) and y = (4, −2, 3, 0), then x + y = (5, 0, 6, 4)

Vector Scaling

• Multiplying a vector by a scalar obtains another vector
• Algebraically, s x = (s x1, s x2,... , s xn) if s is a scalar and x = (x1, x2,... , xn)
• The magnitude (length of arrow) is multiplied by |s|.
• Example: x = (4, −2, 3, 0) and s = (0.5), then sx = (2, −1, 1.5, 0)

Vector Properties

• Commutative: x + y = y + x
• Associative: (x + y) + z = x + (y + z)
• Identity: x + 0 = 0 + x = x, where 0 = (0,... , 0)
• Inverse: x + (−x) = (−x) + x = 0, where −x = (−x1,... , −xn)
• Scalar multiplication identity: 1x = x
• Scalar multiplication associativity: (st)x = s(tx)
• Distributive properties:
  • (s + t)x = sx + tx
  • s(x + y) = sx + sy

Parallel Vectors

• Vectors are if they have the same or opposite direction.

Vector Parametric Equations for Lines

• A line through the origin parallel to (2, 1) means any point has a scalar multiple of d = (2, 1)
• A line can be described as r = s(2,1) where s is an arbitrary scalar

Parametric Equations for Lines Through the Origin

• The equation r = sd represents a line through the origin parallel to vector d

Standard Basis Vectors and Axes

• In 2-space, the x-axis is r = s(1,0) = si and the y-axis is r = s(0,1) = sj

Vector Parametric Equation

• r = r0 + sd where r0 is a known point (position vector) and d is a direction vector, and s is a scalar

Distances and Angles in n-space

• Length in 2-space is measured using Pythagoras' Theorem
• Distance is measured by calculating the length of vector b-a.

Formal Definitions of Lengths, Norms and Magnitudes

• For vector x = (x1,... , xn) in Rn, length formula is ∥x∥ = √(x21 +... + x2n).
• If ∥x∥ = 1, x is a unit vector (x̂).

Distance

• Distance between points x and y is d(x, y) = ∥x − y∥.

Dot Product

• Dot or scalar product or inner product of vectors x = (x1, x2... xn) and y = (y1, y2... yn) is x · y = x1 y1 + x2 y2 + · · · + xn yn
• A scalar not a vector

Dot Product Rules

• Vectors must be the same size
• Order is not important, the product is commutative
• Dot product is distributive with scalar multiplication i.e. s * (x . y) = (s * x) . y = x . (s * y)
• Dot product is distributive with addition i.e. (x + y) . z = (x . z) + (y . z)

Theorems Relating Norms and Dot Products

• ∥x∥2 = x · x
• ∥s x∥ = |s| ∥x∥ for any scalar s ∈ R.
• ∥u + v∥2 + ∥u − v∥2 = 2 ∥u∥2 + 2 ∥v∥2.

Angle Formula

• The angle θ between two non-zero vectors x and y is determined by: cos θ = (x · y) / (∥x∥ ∥y∥)

Orthogonality

• Vectors x and y are orthogonal (or perpendicular) if and only if x · y = 0
• If xy=0, you can write "