Table of Contents
What are Vectors and What is Coplanarity of Vectors?
A vector is a mathematical quantity that has both magnitude and direction. Two vectors are said to be coplanar if they lie in the same plane.
Linearly Dependent and Independent Vectors:
A vector is said to be linearly dependent on a second vector if the first vector can be expressed as a linear combination of the second vector and some other vector(s). A vector is said to be linearly independent of a second vector if the first vector cannot be expressed as a linear combination of the second vector and any other vector(s).
Linearly Independent Vectors:
A vector is linearly independent if it can be expressed as a linear combination of the other vectors in the set, but cannot be expressed as a combination of any other two vectors in the set. In other words, a vector is linearly independent if its components are not all zero.
Linearly Dependent Vectors:
When vectors are linearly dependent, they cannot be represented as a unique linear combination of each other. In other words, at least one of the vectors can be expressed as a linear combination of the other vectors.
For example, the vectors $\vec{a}, \vec{b}, \vec{c}$ are linearly dependent, because the vector $\vec{c}$ can be expressed as a linear combination of the other two vectors:
$$\vec{c} = \vec{a} + \vec{b}$$
On the other hand, the vectors $\vec{a}, \vec{b}, \vec{d}$ are linearly independent, because the vector $\vec{d}$ cannot be expressed as a linear combination of the other two vectors.
Application of Vectors:
There are many different applications for vectors. Some common applications are in physics, engineering, and mathematics.
In physics, vectors are used to describe the motion of objects. In engineering, vectors are used to calculate the forces on objects. In mathematics, vectors are used to solve problems.
Types and Examples of Vector
Fields
A vector field is a mathematical representation of a physical phenomenon in which each point in space is associated with a vector. Vector fields can be used to model a variety of physical phenomena, including electric and magnetic fields, fluid flow, and heat flow.
Some common types of vector fields include:
• Electric field: A vector field that describes the electric force between two points.
• Magnetic field: A vector field that describes the magnetic force between two points.
• Fluid flow: A vector field that describes the motion of a fluid.
• Heat flow: A vector field that describes the flow of heat.
Types of Vectors
There are three types of vectors: magnitude, direction and displacement vectors.
A magnitude vector is a vector that has magnitude only. It has no direction.
A direction vector is a vector that has magnitude and direction.
A displacement vector is a vector that has magnitude and direction. It also has a starting point and an end point.
What are Coplanar Vectors?
Coplanar vectors are two or more vectors that lie in the same plane.
Coplanar Vectors Definition
In mathematics, two vectors are said to be coplanar if they lie in the same plane.
Examples of Coplanar Vectors
A vector is said to be coplanar if it lies in the same plane as some other vector. In the diagram below, vectors A and B are both coplanar.
Vector A is in the same plane as vector B.
Vector A is not in the same plane as vector C.
Points to Look for on Coplanar Vectors:
1. The vectors are perpendicular to each other.
2. The vectors are of the same magnitude.
3. The vectors are of the same direction.
Coplanar Vectors’ Requirements
Two vectors are said to be coplanar if they are both contained within the same plane. In order for two vectors to be coplanar, they must satisfy the following requirements:
The vectors must be linearly independent.
The vectors must be of the same magnitude.
The vectors must be of the same direction.
Uses of Coplanar Vectors
There are several ways to use coplanar vectors. One way is to use them to calculate the magnitude and direction of a resultant vector. Another way is to use them to calculate the magnitude and direction of a force.
