Algebra is a crucial subject in mathematics in which we utilise universal symbols or letters to represent quantities, numbers, and variables. Vector Algebra is a fundamental topic in algebra. It is concerned with the algebra of vector quantities. Scalars and vectors are two sorts of physical quantities, as we all know. A vector quantity has both magnitude and direction, whereas a scalar quantity simply has magnitude.
We execute algebraic operations on vectors and vector spaces. This section contains rules and hypotheses based on vector attributes and behaviour. Here, you will learn several basic algebra concepts including its types like unit vector with examples for better understanding.
Vector Alegbra Definition
A vector is a two-dimensional entity with magnitude and direction. It is typically depicted with an arrow indicating the direction(→) and its length indicating the magnitude. It’s represented by a letter with an arrow on top.
The vector’s magnitude is represented by the symbol |V|.
Vector Algebra Operations
We also execute arithmetic operations on vectors, such as addition, subtraction, and multiplication, just like in regular Algebra. Vectors, however, have two terms for multiplication: dot product and cross product.
Important Operations on Vectors
Consider vector operations like addition, subtraction, and multiplication as follow.
Addition of Vectors
The sum of the two vectors a and b is a + b. This demands joining them from head to tail.
a+b=(a1î+b1ĵ+c1k̂)+(a2î+b2ĵ+c2k̂)=(a1+a2)î+(b1+b2)ĵ+(c1+c2)k̂
Subtraction of Vectors
It is critical to grasp the reverse vector before beginning the method (-a).
A vector has the same magnitude as a reverse vector (-a), but it points in the opposite direction.
First, we calculate the inverse vector.
Then proceed with the addition as usual.
Suppose we want to find vector b – a.
Then, b – a = b + (-a)
Scalar Multiplication of Vectors
The multiplication of a vector by a scalar quantity is known as scaling. In this type of multiplication, just the magnitude of a vector is changed, not its direction.
Cross Product of Vectors
A vector quantity is obtained by taking the cross-product of two vectors. A cross sign between two vectors represents it.
a × b
The mathematical value of a cross product-
a × b= |a||b|sin θ n̂
Types of Vectors – Vector Algebra
Let us explore a few types of vectors based on their magnitude, direction, and their relationship with other vectors.
Zero Vectors
Zero vectors are vectors with a magnitude of zero, denoted by
→
0 = (0,0,0). There are no magnitudes and no directions in the zero vector. It is also known as vector additive identity.
Unit Vectors
Unit vectors are vectors with a magnitude of 1 and are denoted by â.
It is also known as vector multiplicative identity. A unit vector’s magnitude is 1.It’s most commonly used to indicate a vector’s direction.
Position Vectors
Position vectors are utilised in three-dimensional space to calculate the position and direction of movement of vectors. The magnitude and direction of position vectors can be changed in reference to other bodies. It’s also known as the location vector.
Equal Vectors
If the equivalent components of two or more vectors are identical, they are said to be equal. Equal vectors are ones whose magnitude and direction are the same. They can have distinct starting and ending sites, but their magnitude and direction must be the same.
Negative Vector
The inverse of two vectors is when their magnitudes are the same but their directions are opposite. Vector A is said to be the inverse of vector B if their magnitudes are equal but their directions are opposite, and vice versa.
Orthogonal Vectors
When two or more vectors in space form a 90-degree angle, they are said to be orthogonal. In other words, the dot product of orthogonal vectors is always 0. a·b = |a|·|b|cos90° = 0.
Co-initial Vectors
Co-initial vectors are vectors that have the same initial point.
