Vector Operations
Introduction
In mathematics, vectors are objects that have both magnitude and direction. They are commonly used to represent physical quantities like force, velocity, and displacement. Vector operations refer to various mathematical operations that can be performed on vectors, such as addition, subtraction, scalar multiplication, dot product, and cross product.
Definitions and Operations
Addition
Let a = (a₁, a₂, …, aₙ) and b = (b₁, b₂, …, bₙ) be two vectors in n-dimensional space. The sum of these vectors, denoted as a + b, is given by: a + b = (a₁ + b₁, a₂ + b₂, …, aₙ + bₙ)
Subtraction
Similarly, the difference of vectors a and b, denoted as a - b, is given by: a - b = (a₁ - b₁, a₂ - b₂, …, aₙ - bₙ)
Scalar Multiplication
When a vector a is multiplied by a scalar k, the result is a new vector denoted as ka, given by: ka = (ka₁, ka₂, …, kaₙ)
Dot Product
The dot product of two vectors a and b is a scalar quantity denoted as a ⋅ b, and is calculated as: a ⋅ b = a₁b₁ + a₂b₂ + … + aₙbₙ
Cross Product
The cross product of two vectors a and b results in a new vector denoted as a x b, and is given by: a x b = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁)
Real-life Examples
- Addition: Suppose you are walking 3 meters east and then 4 meters north from a point. The combined displacement can be represented by vector addition.
- Scalar Multiplication: If a force vector represents 5 Newtons to the right, 2 times this vector represents 10 Newtons to the right.
- Dot Product: When calculating work done by a force through a distance, the dot product of force and displacement vectors is used.
- Cross Product: In physics, torque is calculated using the cross product of force and lever arm vectors.
Historical Context
Vector operations have been instrumental in various fields, including physics, engineering, computer science, and economics. The concept of vectors was introduced by Josiah Willard Gibbs and Oliver Heaviside in the 19th century, as a way to simplify the study of electromagnetic fields.
Exam Questions
- Calculate the dot product of vectors a = (1, 2, 3) and b = (4, 5, 6).
- Given vectors u = (2, -3) and v = (-1, 4), find the sum u + v and the difference u - v.
- If c = 3a - 2b, find the vector c given a = (2, 1) and b = (-1, 3).