Vector Algebra

A vector is an entity with two dimensions having both a magnitude and a direction. A vector can be visualized geometrically as a directed line segment with an arrow indicating the vector’s direction and a length equal to the vector’s magnitude. The starting point of a vector is called the tail, whereas the endpoint is called the head. We will also understand that they can be displacement vectors, which are the vectors that connect two points (A and B) and position vectors. 

What Is Vector Algebra?

Many algebraic operations involving vectors are performed using vector algebra. The magnitude of the vector is the length of the line between two points A and B, and the direction of the vector AB is the direction of displacement of point A to point B. There are two types of vector algebra: displacement vector and position vector.

A position vector is a vector that represents the position or location of any given point in relation to any ambiguous reference point such as the origin.

The displacement vector is the switch in the position vector of an object.

Algebraic Vectors in the Coordinate System

Algebraic vectors are related to a coordinate system. For vector V→= (2,3), (2,3) is the point located in the coordinate system, where 2 corresponds to x and 3 to y.

Representation of Vectors

Vectors are usually written in bold lowercase letters, like b, or with an arrow over the letter, like b→. 

Vectors can also be denoted by an arrow above their initial and terminal points. For example, vector BC can be denoted as BC→ . A = ci + dj+ ek is the usual form of representation for vectors. Here, c,d,e are real numbers, and I, j, and k are the unit vectors along the x, y, and z axes, respectively. The unit vectors have been explained below.

Unit Vector and Direction of the Vector

Unit vectors are defined as vectors with a magnitude equal to one and are denoted by the letter  a. 

Unit vectors have a length of one. Unit vectors indicate a vector’s direction.

The direction of a unit vector is the same as that of the specified vector, but its magnitude is one unit. A unit vector for a vector B is

B =B/ | B | 

Vectors’ Magnitude

A vector’s magnitude is a scalar value.

The square root of the sum of the squares of a vector’s elements can be used to calculate its magnitude. If the components of a vector B are (x,y,z), then the magnitude formula for B is

|B| = x2+y2+z2                                

Components of Vector

In a two-dimensional coordinate system, any algebraic vector can be broken down into its x- and y-components. 

Since the formula gives the magnitude of any vector:

|V| = √[Vx2 + Vy2], where Vx and Vy are the components of vector V in the x-axis and y-axis, respectively.

The individual components can be calculated using the following formula:

Vx = V·cosθ, and Vy = V.sinθ

Angle Between Two Vectors

The angle between two vectors is calculated using the following formula:

θ = cos-1[(a·b)/|a||b|], where θ is the angle between two vectors, a and b are considered, and |a| and |b| are their magnitudes respectively.

Operations of Vector

Addition of Vectors

The process of adding algebraic vectors is identical to that of adding scalars. Both commutative and associative rules hold for adding vectors.

To reach the final result, the constituent components of the respective vectors are added. Mathematically, it is represented as:

c + d = (k1 i + l1j+ m1 k) + (k2i + l2j+ m2 k)

= (k1, l1, m1) + (k2, l2, m2)

= (k1 + l2, m1 + k2, l1 + m2)

= (k1 + k2)  i + (l1 + l2)j + (m1 + m2) k

Subtraction of Vectors

In the case of subtraction, just the direction of one of the vectors is modified, and the other vector is added to it.

a – b = (k1i + l1j + m1 k) – (k2i + l2j+m2 k)

= (k1, l1, m1) – (k2, l2, m2)

= (k1 – k2, l1 – l2, m1 – m2)

=(k1 – k2)i + (l1 – l2)j+ (m1 – m2) k

Multiplication of Vectors

The multiplication algorithms of algebraic vectors differ slightly from those of real numbers. There are two different methods for multiplying vectors:

• The Dot Product of Vectors

To get the dot product of two vectors, multiply the individual components of the two vectors to be multiplied and add the result.

a·b = (a1i + b1j + c1  k) ·(a2 i + b2 j+ c2 k)

= (a1, b1, c1)·(a2, b2, c2)

= (a1·a2) + (b1·b2) + (c1·c2)

Another technique to find the dot product of two algebraic vectors C and D is by calculating

using the following formula:

C⋅ D = |C||D| cosθ

• The Cross Product of Vector

The cross product of two algebraic vectors, A and B, is evaluated by the product of the magnitudes of the two vectors and the sine of the angle between them.

A× B = |A||B| sinθ n

Conclusion

A magnitude and a direction are the two fundamental features of an algebraic vector. When two vectors have the same magnitude and direction, they are equal. Many physical quantities, such as velocity, displacement, acceleration, and force, are vector values, meaning they have both a magnitude and a direction.