Vector Math

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Vector Math

Last Update: June 20, 2025

Introduction
Vectors and Scalars
Vector Addition
Vector Subtraction
Vector Multiplication by a Scalar
Length (Magnitude) of a Vector
Vector Dot Product
Vector Cross Product

Introduction

Vectors are mathematical constructs that indicate magnitude and time.

They are very useful in science and engineering fields.

In this post, we’ll cover the basic math of vectors.

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Vectors and Scalars

Scalar

A scalar only has magnitude (size).

Example: Speed

Vector

A vector has magnitude (size) and direction.

Example: Velocity = Speed in a specified direction

Example: Force has magnitude and direction

Adding/Subtracting Vectors

Vectors have magnitude and direction but graphically they don’t specifically place anywhere. i.e. they can be placed anywhere.

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Vector Addition

To add vectors algebraically, sum their corresponding components.

If a $= (a_{x},a_{y},a_{z})$ and b $= (b_{x},b_{y},b_{z})$ , then: a $+b = (a_{x} +b_{x},a_{y} +b_{y},a_{z} +b_{z})$

Vector Addition: Graphical (Tip-to-Tail / Triangle Rule):

Place the tail of the second vector (b) at the tip (arrowhead) of the first vector (a).
The resultant sum vector (a+b) is drawn from the tail of the first vector to the tip of the second vector.

Vector Addition: Graphical (Parallelogram Rule):

Draw both vectors (a and b) with their tails at the same starting point.
Complete the parallelogram formed by these two vectors as adjacent sides.
The resultant sum vector (a+b) is the diagonal of the parallelogram that starts from the common tail point.

Properties of Vector Addition:

For vectors a, b, c and the zero vector $0$ :

Commutativity: The order of addition does not affect the sum. a + b = b + a
Associativity: The grouping of vectors in addition does not affect the sum. $(a + b) + c = a + (b + c)$
Identity Element (Zero Vector): Adding the zero vector to any vector leaves the vector unchanged: a $+ 0 = a$
Inverse Element (Negative Vector): For every vector a, there exists a unique negative vector $(-a)$ such that their sum is the zero vector: a $+ (-a) = 0$
Closure: The sum of two vectors is always another vector.

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Vector Subtraction

Subtracting vector b from vector a, (a $-b$ ), is defined as adding vector a to the negative of vector b: (a $+ (-b)$ ).
The negative vector $(-b)$ has the same magnitude as b but points in the exact opposite direction (i.e. flipped 180 degrees).
Subtract the corresponding components of the vectors.
- If a $= (a_{x},a_{y},a_{z})$ and b $= (b_{x}, b_{y}, b_{z})$ , then: a $- b = (a_{x} - b_{x},a_{y} - b_{y},a_{z} - b_{z})$
Graphical (tip to tip): If vectors a and b are drawn with their tails at the same starting point, the resultant vector a $-b$ is drawn from the tip of b to the tip of $a$ .

Graphical (flip subtrahend vector and add): Flip the subtrahend vector 180 degrees and add this to the first vector

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Vector Multiplication by a scalar

Multiplying a vector by a positive scalar increases the magnitude of the vector
Multiplying a vector by negative one, flips the direction by 180 degrees

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Length (Magnitude of a vector)

The length or magnitude of a vector is a scalar quantity that represents the “size” or “extent” of the vector, without regard to its direction.

The magnitude of a vector $v$ is typically denoted as $∥ v ∥$ or $∣ v ∣$ .

For a vector in a coordinate system:

In two dimensions (e.g., $v = ⟨ x, y ⟩$ ): The magnitude is calculated using the Pythagorean theorem: $∥ v ∥ =sqrt(x 2 + y 2$
In three dimensions (e.g., $v = ⟨ x, y, z ⟩$ ): The magnitude is also found using a generalization of the Pythagorean theorem: $∥ v ∥ =sqrt(x 2 + y 2 + z 2$

This makes sense. Just plot

This concept extends to vectors in any number of dimensions, where the magnitude is the square root of the sum of the squares of all its components.

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Vector Dot Product

The dot product (also known as the scalar product or inner product) of two vectors is an algebraic operation that takes two equal-length sequences of numbers (vectors) and returns a single number (a scalar).

Vector Dot Product Algebraic Definition

Given two vectors $a = ⟨ a_{1}, a_{2}, \dots, a_{n} ⟩$ and $b = ⟨ b_{1}, b_{2}, \dots, b_{n} ⟩$ in $n$ -dimensional space,

their dot product is the sum of the products of their corresponding components:
$that is, a \cdot b = a_{1} b_{1} + a_{2} b_{2} + \dots + a_{n} b_{n} = \sum a_{i} b_{i for i = 1 to n}$

Vector Dot Product Geometric Definition

Given two vectors $a$ and $b$ in Euclidean space, their dot product is the product of their magnitudes and the cosine of the angle $θ$ between them:

where is the magnitude (length) of vector ,
and is the magnitude of vector .
Lower case versus bold is sometimes used to differentiate a vector from its scalar form.
The Dot product a · b
- = scalar product of b & “the projection of a onto b” = bacos(θ)
- = scalar product of a & “the projection of b onto a” = abcos(θ)
The dot product is the product of the lengths of the vectors that are aligned in the same direction.
The maximum value of a dot product will occur when the vectors are colinear (aligned the same way i.e. θ =0)
When vectors a and b are perpendicular to each other, their dot product is zero (i.e. θ =90 degrees).

Picture: Geometric Interpretation of Dot Product

Properties of the Dot Product:

Let $u$ , $v$ , and $w$ be vectors, and $c$ be a scalar.

Commutative Property: The order of the vectors does not affect the result.
Distributive Property over Vector Addition:
Scalar Multiplication Property (Associative with Scalar Multiplication):
Relationship with Magnitude (Squared Length): The dot product of a vector with itself gives the square of its magnitude. .This also implies that .
Orthogonality Condition: Two non-zero vectors are perpendicular (orthogonal) if and only if their dot product is zero. (for non-zero and )
Dot Product with the Zero Vector: The dot product of any vector with the zero vector is zero.

These properties make the dot product a fundamental tool in various areas of mathematics, physics, and engineering, particularly for calculating angles, projections, and work done by forces.

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Vector Cross Product

The cross product is defined only in three-dimensional Euclidean space. (each vector can only have three components)
Unlike the dot product, the result of a cross product is another vector, not a scalar and it is orthogonal to the the multiplicand vectors

It is denoted as $a \times b$ .

The cross product $a \times b$ is a vector $c$ that possesses the following characteristics:

Direction: The vector $c$ is orthogonal to both $a$ and b (if the vectors are non zero vectors we can say perpendicular)

Picture: 3 D Coordinates of a Cross Product (order of first and second vectors are unique)

Geometrically the cross product of vectors a and b is a measure of how orthogonal (perpendicular for non zero vectors) a and b are to each other.
The cross product magnitude is a maximum when the angle between the vectors is 90 degrees.
The cross product is a vector whose direction will be orthogonal to both vectors a and b\
The magnitude of $c = a x b$ is given by the formula: $| c | = |axb| =| a| | b| sin (θ)$ where $θ$ is the angle between $a$ and $b$ ( $0 \leq θ \leq π$ ).
Vector Form: a x b = |a||b|sin(θ)n = absin(θ)n = basin(θ)n
Scalar Form: |a x b| = |a||b|sin(θ) = absin(θ) = basin(θ)
- where
- | | indicates magnitude of vector or vector product (or the scalar form is sometimes just un-bolded)
- n is the unit normal vector
Geometrically, this magnitude represents the area of the parallelogram formed by vectors $a$ and $b$ when their tails are at the same point.

Picture: Geometric Interpretation of Cross Product

a x b does not equal b x a
The direction of the cross product is determined by the right-hand rule
- If you curl the fingers of your right hand from the direction of $a$ to the direction of $b$ (through the smaller angle between them), your thumb will point in the direction of $c$ .
There are actually two other right hand rule methods you can use. I show schematics for all three methods below.

Method 1: Right Hand: Put open hand on plane of vector a, then curl fingers towards b, thumb points in direction of c.

Picture: Vector Cross Product Right Hand Rule Method 1 (Curl)

Method 2: Right Hand: Index towards a, Middle finger towards b, thumb points in direction of c

Picture: Vector Cross Product Right Hand Rule Method 2 (Index, Middle, Thumb)

Method 3: Right Hand. Open hand. Thumb towards a and other fingers toward b. Then c comes out of open side of palm.

Picture: Vector Cross Product Right Hand Rule Method 3 (Open Palm)

Algebraic Definition (in Cartesian Coordinates)

Let a = three dimensional vector a and

let b = three dimensional vector b

a = (ax, ay, az)

b = (bx, by, bz)

c = a x b = a vector with three components that are computed as the determinants of three 2 x 2 matrices:

If you look at the a and b vectors as a collection of three columns of components: col1= ax/bx, col2 = ay/by and col3 = az/bz,

notice that the sequence of matrices to compute the determinant on are col2 with col3, then col3 with col1, then col 1 with col2 (a left to right sequence).

So,

c = a x b = (aybz – byaz, azbx – bzax, axby – bxay)

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Menu (linked Index)

Introduction

Vectors and Scalars

Scalar

Vector

Adding/Subtracting Vectors

Vector Addition

Vector Addition: Graphical (Tip-to-Tail / Triangle Rule):

Vector Addition: Graphical (Parallelogram Rule):

Properties of Vector Addition:

Vector Subtraction

Vector Multiplication by a scalar

Length (Magnitude of a vector)

Vector Dot Product

Vector Dot Product Algebraic Definition

$that is, a \cdot b = a_{1} b_{1} + a_{2} b_{2} + \dots + a_{n} b_{n} = \sum a_{i} b_{i for i = 1 to n}$

Vector Dot Product Geometric Definition

The dot product is the product of the lengths of the vectors that are aligned in the same direction.

Picture: Geometric Interpretation of Dot Product

Properties of the Dot Product:

Vector Cross Product

The cross product $a \times b$ is a vector $c$ that possesses the following characteristics:

Picture: 3 D Coordinates of a Cross Product (order of first and second vectors are unique)

Geometrically the cross product of vectors a and b is a measure of how orthogonal (perpendicular for non zero vectors) a and b are to each other.

The magnitude of $c = a x b$ is given by the formula: $| c | = |axb| =| a| | b| sin (θ)$ where $θ$ is the angle between $a$ and $b$ ( $0 \leq θ \leq π$ ).

Picture: Geometric Interpretation of Cross Product

Picture: Vector Cross Product Right Hand Rule Method 1 (Curl)

Picture: Vector Cross Product Right Hand Rule Method 2 (Index, Middle, Thumb)

Picture: Vector Cross Product Right Hand Rule Method 3 (Open Palm)

Algebraic Definition (in Cartesian Coordinates)

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Menu (linked Index)

Introduction

Vectors and Scalars

Scalar

Vector

Adding/Subtracting Vectors

Vector Addition

Vector Addition: Graphical (Tip-to-Tail / Triangle Rule):

Vector Addition: Graphical (Parallelogram Rule):

Properties of Vector Addition:

Vector Subtraction

Vector Multiplication by a scalar

<img loading="lazy" loading="lazy" decoding="async" class="alignnone size-full wp-image-31416" src="https://wymhacks.com/wp-content/uploads/2025/07/2025.04.21-graph-vector-multiplication-by-a-scalar.jpg" alt="" width="403" height="539" />

Length (Magnitude of a vector)

Vector Dot Product

Vector Dot Product Algebraic Definition

that is, a⋅b=a1b1+a2b2+⋯+anbn=∑aibi for i = 1 to n

Vector Dot Product Geometric Definition

a⋅b=∣a∣∣b∣cos(θ)

The dot product is the product of the lengths of the vectors that are aligned in the same direction.

Picture: Geometric Interpretation of Dot Product

Properties of the Dot Product:

Vector Cross Product

The cross product a×b is a vector c that possesses the following characteristics:

Picture: 3 D Coordinates of a Cross Product (order of first and second vectors are unique)

<img loading="lazy" loading="lazy" decoding="async" class="alignnone size-full wp-image-31428" src="https://wymhacks.com/wp-content/uploads/2025/07/2025.04.21-Vector-cross-product-graph-1.jpg" alt="" width="428" height="841">

Geometrically the cross product of vectors a and b is a measure of how orthogonal (perpendicular for non zero vectors) a and b are to each other.

The magnitude of c = a x b is given by the formula: |c|= |axb| =|a||b|sin(θ) where θ is the angle between a and b (0≤θ≤π).

Picture: Geometric Interpretation of Cross Product

<img loading="lazy" loading="lazy" decoding="async" class="alignnone size-full wp-image-31417" src="https://wymhacks.com/wp-content/uploads/2025/07/2025.04.21-graph-vector-cross-product.jpg" alt="" width="612" height="743">

Picture: Vector Cross Product Right Hand Rule Method 1 (Curl)

Picture: Vector Cross Product Right Hand Rule Method 2 (Index, Middle, Thumb)

Picture: Vector Cross Product Right Hand Rule Method 3 (Open Palm)

Algebraic Definition (in Cartesian Coordinates)

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$that is, a \cdot b = a_{1} b_{1} + a_{2} b_{2} + \dots + a_{n} b_{n} = \sum a_{i} b_{i for i = 1 to n}$

The cross product $a \times b$ is a vector $c$ that possesses the following characteristics:

The magnitude of $c = a x b$ is given by the formula: $| c | = |axb| =| a| | b| sin (θ)$ where $θ$ is the angle between $a$ and $b$ ( $0 \leq θ \leq π$ ).