Vectors
Understanding Vectors
Vectors are fundamental in physics and mathematics as they describe quantities that have both magnitude and direction. They are different from scalars, which have only magnitude. Vectors are used to represent physical quantities such as displacement, velocity, acceleration, and force.
Definition of a Vector
A vector is a mathematical object characterized by:
- Magnitude: The size or length of the vector.
- Direction: The orientation of the vector in space.
Vectors are usually denoted by boldface letters (e.g., v) or letters with an arrow on top (e.g., $\vec{v}$).
Representation of Vectors
Vectors can be represented in several ways:
- Graphically: As an arrow with a direction and length proportional to the vector's magnitude.
- Component Form: Using coordinates, such as $(x, y)$ in 2D or $(x, y, z)$ in 3D.
- Unit Vector Notation: A vector can be expressed as the product of its magnitude and a unit vector (a vector with a magnitude of 1) in the direction of the vector.
Operations with Vectors
Vectors can be added, subtracted, and multiplied by scalars. Here are some of the basic operations:
Vector Addition
Vectors are added tip-to-tail graphically, or by adding their corresponding components algebraically.
$$ \vec{a} + \vec{b} = (a_x + b_x, a_y + b_y, a_z + b_z) $$
Vector Subtraction
Subtraction is similar to addition but involves subtracting the corresponding components.
$$ \vec{a} - \vec{b} = (a_x - b_x, a_y - b_y, a_z - b_z) $$
Scalar Multiplication
Multiplying a vector by a scalar changes the magnitude of the vector but not its direction.
$$ k\vec{a} = (ka_x, ka_y, ka_z) $$
where $k$ is a scalar.
Dot Product
The dot product (or scalar product) is an operation that takes two vectors and returns a scalar.
$$ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) = a_x b_x + a_y b_y + a_z b_z $$
where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$.
Cross Product
The cross product (or vector product) is an operation that takes two vectors in three-dimensional space and returns a vector that is perpendicular to both.
$$ \vec{a} \times \vec{b} = (a_y b_z - a_z b_y, a_z b_x - a_x b_z, a_x b_y - a_y b_x) $$
Differences Between Scalars and Vectors
| Property | Scalars | Vectors |
|---|---|---|
| Magnitude | Only magnitude | Magnitude and direction |
| Examples | Mass, time | Force, velocity |
| Operations | Addition, multiplication, division | Addition, subtraction, scalar multiplication, dot product, cross product |
| Representation | Real numbers | Arrows, ordered pairs/triples, unit vector notation |
Examples
Example 1: Vector Addition
If $\vec{a} = (3, 4)$ and $\vec{b} = (1, 2)$, then the sum $\vec{a} + \vec{b}$ is:
$$ \vec{a} + \vec{b} = (3 + 1, 4 + 2) = (4, 6) $$
Example 2: Dot Product
Given $\vec{a} = (1, 2, 3)$ and $\vec{b} = (4, -5, 6)$, the dot product $\vec{a} \cdot \vec{b}$ is:
$$ \vec{a} \cdot \vec{b} = 1 \cdot 4 + 2 \cdot (-5) + 3 \cdot 6 = 4 - 10 + 18 = 12 $$
Example 3: Cross Product
For $\vec{a} = (1, 0, 0)$ and $\vec{b} = (0, 1, 0)$, the cross product $\vec{a} \times \vec{b}$ is:
$$ \vec{a} \times \vec{b} = (0 \cdot 0 - 0 \cdot 1, 0 \cdot 0 - 0 \cdot 0, 1 \cdot 1 - 0 \cdot 0) = (0, 0, 1) $$
This result is a vector pointing in the direction of the z-axis, which is perpendicular to both $\vec{a}$ and $\vec{b}$.
Vectors are a powerful tool in physics and mathematics, providing a way to describe and calculate properties of physical systems. Understanding how to work with vectors is essential for solving problems in these fields.