Contents
This article explains vectors, which are mathematical objects defined by both a magnitude and a direction, exploring their representation, types, and operations in physics and engineering. I will now explore these essential mathematical tools with you. A geometric object that possesses both magnitude and a specific direction is a fundamental quantity we will examine. These are essential for representing physical phenomena like force and displacement. The modern use of these mathematical objects began in the late 19th century, evolving from the work of mathematicians like William Rowan Hamilton and Hermann Grassmann.
Core Definitions
The word itself comes from Latin, meaning “carrier”. The magnitude, or norm, of a geometric object is a scalar quantity that represents its length. Its direction specifies the orientation in space, often indicated by an arrow. These quantities are sometimes referred to as Euclidean vectors in geometry. In mathematics, we define them as elements of a specific algebraic structure. The initial point of a representation is where it begins, while the terminal point marks where it ends.
Applications and Examples
I find these concepts have broad applications in physics, engineering, and computer graphics. Some physical quantities represented by these mathematical objects include velocity, acceleration, and magnetic fields. A quantity defined by both a magnitude and a direction is a vector.
Vector Representation and Core Properties
Understanding how to represent these quantities is the first step toward using them effectively.
Representing a Vector
We typically denote these quantities with an arrow over a letter, like $\vec{a}$, or by using boldface type, such as a. The standard form represents them as an ordered list of components, such as <x, y, z>. The initial and terminal points are also called the tail and head, respectively.
In a Cartesian coordinate system, a two-dimensional object can be denoted as v = (vx, vy), where vx and vy are its horizontal and vertical components.
[Diagram: A Cartesian plane showing a vector from the origin (0,0) to a point (3,4), labeled as vector A. The horizontal component (3) and vertical component (4) are shown, forming a right-angled triangle with the vector as the hypotenuse.]
Magnitude and Angle
1. Magnitude
You can calculate the magnitude using the Pythagorean theorem on its components. The formula for the magnitude of a two-dimensional vector v = (v_x, v_y) is $|\mathbf{v}| = \sqrt{v_x^2 + v_y^2}$. The magnitude is always a non-negative scalar quantity, never a vector itself.
2. Angle Between Vectors
Calculating the angle between two of these objects helps determine their orientation relative to each other. This angle indicates if they are pointing in similar, opposite, or perpendicular directions. We use the dot product formula to find this angle: $\theta = \arccos\left(\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}\right)$.
For a great visual explanation of these concepts, I recommend this video:
Types of Vectors
I will now explain how different types of these mathematical quantities are termed based on their distinct properties.
- Zero Vectors: A zero vector, denoted as $\vec{0}$ or 0, has zero magnitude and an undefined direction, and it acts as the additive identity in vector spaces.
- Unit Vectors: These are objects with a magnitude of exactly one, which are generally used to specify a direction without altering magnitude in calculations.
- Position Vectors: We use position vectors, also known as radius vectors, to specify the location of a point in space relative to an arbitrary origin.
- Equal Vectors: Two or more of these quantities are equal if they share the exact same magnitude and the identical direction, regardless of their initial points.
- Negative Vector: The negative of a vector v is a quantity that has the same magnitude as v but points in the precise opposite direction.
- Parallel & Antiparallel Vectors: Parallel quantities point in the same direction (angle of 0°), while antiparallel ones point in opposite directions (angle of 180°).
- Orthogonal Vectors: Two objects in space are orthogonal if the angle between them is 90 degrees, meaning they are perpendicular to one another. The dot product of orthogonal quantities is always zero.
- Co-initial Vectors: These are multiple quantities that share the same starting point or origin.
Vector Operations, Formulas, and Properties
Here, I will detail the fundamental operations and formulas that govern these quantities.
Summary of Vector Formulas
I have compiled a table to summarize the key formulas for quick reference.
| Operation | Formula | Result Type |
|---|---|---|
| Addition | a + b = (a_x + b_x, a_y + b_y) | Vector |
| Subtraction | a – b = (a_x – b_x, a_y – b_y) | Vector |
| Dot Product | a · b = a_xb_x + a_yb_y | Scalar |
| Cross Product (3D) | a × b = (a_yb_z – a_zb_y, a_zb_x – a_xb_z, a_xb_y – a_yb_x) | Vector |
| Angle Between Two | $\theta = \arccos\left(\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|}\right)$ | Scalar |
Detailed Operations
The basic algebraic operations are addition, subtraction, and two different forms of multiplication.
1. Addition of Vectors
We add these quantities by summing their corresponding components. This can be visualized using the Triangle Law of Addition, where you place the tail of the second object at the head of the first. The Parallelogram Law of Addition provides an alternative visualization where the resultant is the diagonal of a parallelogram formed by the two initial objects.
2. Subtraction of Vectors
Subtracting one quantity from another is equivalent to adding its negative counterpart.
3. Scalar Multiplication
We can multiply a vector by a scalar, which is an operation called scaling. This process changes the magnitude of the object; its direction remains the same unless multiplied by a negative scalar, which reverses it.
4. Vector Multiplication
I want to emphasize that there are two distinct ways to multiply these quantities together.
- Dot Product: The dot product, or scalar product, is calculated by multiplying corresponding components and summing the results, yielding a single scalar value.
- Cross Product: The cross product, or vector product, is calculated for three-dimensional objects and results in a new vector that is perpendicular to the plane containing the original two.
Vectors vs. Scalars
It is crucial to distinguish between scalar and vector quantities.
Distinguishing Scalars and Vectors
A quantity defined by both a magnitude and a direction is a vector. Scalars are quantities that are fully described by a magnitude alone. The primary difference is the inclusion of directional information for vector quantities.
| Feature | Scalar | Vector |
|---|---|---|
| Definition | A quantity with only magnitude. | A quantity with both magnitude and direction. |
| Examples | Speed, mass, temperature, distance, time | Velocity, force, acceleration, displacement |
| Notation | Regular Italic (e.g., m for mass) | Boldface (v) or Arrow ($\vec{v}$) |
| Addition | Simple arithmetic | Parallelogram or Triangle Law |
Summary and Frequently Asked Questions (FAQs)
Let’s conclude with a look at real-world uses and some common questions.
Real-World Applications
I see applications of these concepts everywhere, from air traffic control systems that manage aircraft velocity and position, to computer-generated imagery (CGI) where they define object orientation, to physics calculations that determine the effect of multiple forces on a body.
FAQs
What are Vectors in Math?
In mathematics, they are elements of a vector space, which are geometric objects characterized by having both a magnitude and a direction.
What are Examples of Vectors?
Common examples from physics include displacement, velocity, acceleration, force, and momentum, all of which require both a size and a direction to be fully described.
What are the Properties of Vectors?
Key properties include their ability to be added, subtracted, and multiplied (via dot and cross products), as well as properties of commutativity and associativity for addition.
What are Collinear Vectors?
Collinear quantities are those that lie along the same line or are parallel to each other, meaning they can be expressed as scalar multiples of one another.
