Fundamental Theorem Of Vectors In 2D and 3D

Vectors are important mathematical tools that are utilised in many disciplines, including physics, engineering, and computer science. They are useful in representing physical quantities such as displacement, velocity, and force since they describe quantities with both magnitude and direction. The Basic Theorem of Vectors is a set of ideas that aids in the understanding and manipulation of vectors.

Table of Contents:

What is the Fundamental Theorem of Vectors?
Types of Vectors
Fundamental Theorem of Vectors in Two-Dimensions
Fundamental Theorem of Vectors in Three-Dimensions

What is the Fundamental Theorem of Vectors?

The Fundamental Theorem of Vectors establishes the link between vectors and their components in linear algebra. It asserts that each vector in a vector space may be written uniquely as a linear combination of a collection of linearly independent vectors. To put it another way, it enables us to express any vector in terms of its components along a collection of basis vectors.

Types of Vectors

Before delving into the Fundamental Theorem of Vectors, it is necessary to first grasp the many forms of vectors.

Row Vectors: Row vectors are vectors that are expressed horizontally and have a single row of items. [2, 4, 6] is an example of a row vector.
Column Vectors: Column vectors are vectors that are expressed vertically and have a single column of items. [2, 4, 6] is an example of a column vector.
Null Vectors: Null vectors, often known as zero vectors, are vectors with no magnitude or direction. The vector [0, 0, 0] represents them.
Unit Vectors: Vectors with one magnitude are known as unit vectors. They are frequently used in coordinate systems to define directions. In three-dimensional space, examples are the unit vectors I, j, and k.

Fundamental Theorem of Vectors in Two-Dimensions

If a and b are non-zero non-collinear vectors, then every vector r in their plane may be uniquely written as a linear combination of a and b, i.e. there exist unique l, m R such that

la + mb = r.
That is, if l1a + m1b = l2a + m2b,

then l1 = l2 and m1 = m2.

Fundamental Theorem of Vectors in Three-Dimensions

Any vector r in space may be uniquely expressed as a linear combination of a, b, and c if a, b, and c are three non-zero, non-coplanar vectors in space. There exist distinct l, m, n R such that

la + mb + nc = r.
If l1a + m1b + n1c = l2a + m2b + n2c,

then l1 equals l2, m1 equals m2, and n1 equals n2.

Practice Problems

Q1. Which vector operation attribute allows us to prioritise scalar multiplication over vector addition?

a. Associative Property of Vector Addition
b. Distributive Property of Scalar Multiplication
c. Commutative Property of Vector Addition
d. Identity Property of Vector Addition

Ans: b. Distributive Property of Scalar Multiplication

The distributive property asserts that we may distribute scalar multiplication across vector addition for any scalar 'a' and vectors 'u' and 'v' as (a(u + v)) = (au + av).

Q2. What are the several types of vectors that have a magnitude of one and are used to describe directions?

a. Null Vectors
b. Row Vectors
c. Unit Vectors
d. Column Vectors

Ans: c. Unit Vectors

Unit vectors are vectors with a magnitude of one that is often used in coordinate systems to denote directions.

Q3. According to which theory any vector may be represented as a linear combination of linearly independent basis vectors?

a. Fundamental Theorem of Vectors
b. Pythagorean Theorem
c. Central Limit Theorem
d. Law of Sines

Ans: a. Fundamental Theorem of Vectors
Each vector in a vector space may be uniquely represented as a linear combination of linearly independent basis vectors, according to the Basic Theorem of Vectors.

Frequently Asked Questions

Q1. Vectors may be added or subtracted.
Answer: Vectors may be added and removed using vector addition and vector subtraction procedures.

Q2. How may the Basic Theorem of Vectors be implemented in practice?
Answer: The Basic Theorem of Vectors is frequently used to represent and evaluate numerous physical events and systems in physics, engineering, and computer graphics.

Q3. Can the Vector Basic Theorem be extended to greater dimensions?
Answer: The Vector Basic Theorem is limited to three dimensions but cannot directly extend to higher dimensions. There are, however, similar theorems for higher-dimensional spaces. Stokes' theorem extends the Vector Basic Theorem to higher dimensions. Stokes' theorem connects integrals over a higher-dimensional manifold (such as a surface or a volume) to its boundary. It extends the Vector Basic Theorem by linking the theories of integration and differentiation in several dimensions.