Vector Algebra

Vectors algebra involves algebraic operations across vectors. The algebraic operations involving the magnitude and direction of vectors is performed in vector algebra.  Vector algebra helps for numerous applications in physics, and engineering to perform addition and multiplication operations across physical quantities, represented as vectors in three-dimensional space. 

Let us learn more about vector algebra, operations in vector algebra, vector types, with the help of solved examples, and practice questions.

Vector algebra is used to perform numerous algebraic operations involving vectors. A vector is a Latin word that means carrier. Vectors carry a point A to point B. The length of the line between the two points A and B is called the magnitude of the vector and the direction or the displacement of point A to point B is called the direction of the vector AB. Vectors are also called Euclidean vectors or Spatial vectors. Vectors have many applications in maths, physics, engineering, and various other fields.

Vectors in math is a geometric entity that has both magnitude and direction. Vectors have an initial point at the point where they start and a terminal point that tells the final position of the point. Various algebraic operations such as addition, subtraction, and multiplication can be performed in vector algebra. Many of the physical quantities such as velocity, displacement, acceleration, force are all vector quantities that have a magnitude as well as a direction.

Representation of Vectors

Vectors are usually represented in bold lowercase such as a or using an arrow over the letter as \(\vec{a}\). Vectors can also be denoted by their initial and terminal points with an arrow above them, for example, vector AB can be denoted as \(\overrightarrow{AB}\). The standard form of representation of a vectors is \(\vec{A}=a \hat{i}+b\hat{j}+c\hat{k}\). Here, a,b,c are real numbers and \(\hat{i}, \hat{j}, \hat{k}\) are the unit vectors along the x-axis, y-axis, and z-axis respectively.

The initial point of a vector is also called the tail whereas the terminal point is called the head. Vectors describe the movement of an object from one place to another. In the cartesian coordinate system, vectors can be denoted by ordered pairs. Similarly, vectors in 'n' dimensions can be denoted by an 'n' tuple. Vectors are also identified with a tuple of components which are the scalar coefficients for a set of basis vectors. The basis vectors are denoted as: e1 = (1,0,0), e2 = (0,1,0), e3 = (0,0,1)

Magnitude of Vectors

The magnitude of a vector can be calculated by taking the square root of the sum of the squares of its components. If (x,y,z) are the components of a vector A, then the magnitude formula of A is given by,

|A| = √ (x²+y²+z²)

The magnitude of a vector is a scalar value.

Components of Vectors

A vector quantity has two characteristics, magnitude, and direction, and both the quantities are compared while comparing two vector quantities of the same type. Any vector, in a two-dimensional coordinate system, can be broken into x-component, and y-component. In the figure given below, we can observe these components - x-component, V\(_x\) and y-component, V\(_y\) for a vector,v in coordinate plane.

The values of V\(_x\) and V\(_y\) can be given as,

V\(_x\) = V·cosθ, and V\(_y\) = V.sinθ

|V| = √[V\(_x\)² + V\(_y\)²]

Angle Between Two Vectors

The angle between two vectors can be calculated using the dot product formula. Let us consider two vectors a and b and the angle between them to be θ. Then, the dot product of two vectors is given by a·b = |a||b| cosθ. We need to determine the value of the angle θ. The angle between two vectors also indicates the directions of the two vectors. θ can be evaluated using the following formula:

θ = cos^-1[(a·b)/|a||b|]

The vector algebra has different types of vectors that are used for different algebraic. The vectors are termed as different types based on their magnitude, direction, and their relationship with other vectors. Let us explore a few types of vectors and their properties:

Zero Vectors

Vectors that have 0 magnitude are called zero vectors, denoted by \(\overrightarrow{0}\) = (0,0,0). The zero vector has zero magnitudes and no direction. It is also called the additive identity of vectors.

Unit Vectors

Vectors that have magnitude equals to 1 are called unit vectors, denoted by \(\hat{a}\). It is also called the multiplicative identity of vectors. The magnitude of a unit vectors is 1. It is generally used to denote the direction of a vector.

Position Vectors

Position vectors are used to determine the position and direction of movement of the vectors in a three-dimensional space. The magnitude and direction of position vectors can be changed relative to other bodies. It is also called the location vector.

Equal Vectors

Two or more vectors are said to be equal if their corresponding components are equal. Equal vectors have the same magnitude as well as direction. They may have different initial and terminal points but the magnitude and direction must be equal.

Negative Vector

A vector is said to be the negative of another vector if they have the same magnitudes but opposite directions. If vectors A and B have equal magnitude but opposite directions, then vector A is said to be the negative of vector B or vice versa.

Parallel Vectors

Two or more vectors are said to be parallel vectors if they have the same direction but not necessarily the same magnitude. The angles of the direction of parallel vectors differ by zero degrees. The vectors whose angle of direction differs by 180 degrees are called antiparallel vectors, that is, antiparallel vectors have opposite directions.

Orthogonal Vectors

Two or more vectors in space are said to be orthogonal if the angle between them is 90 degrees. In other words, the dot product of orthogonal vectors is always 0. a·b = |a|·|b|cos90° = 0.

Co-initial Vectors

Vectors that have the same initial point are called co-initial vectors.

Some basic operations in vector algebra can be performed geometrically without taking any coordinate system as a reference. These various operations which can be performed on vectors are addition, subtraction, and multiplication by a scalar. Also, the two different ways of multiplication of vectors are the dot product and the cross product of vectors. The different operations in vector algebra are as follows.

• Addition of Vectors
• Subtraction of Vectors
• Scalar Multiplication
• Scalar Triple Product of Vectors
• Multiplication of Vectors

Let us understand each of these operations in vector algebra in the below paragraphs.

Addition of Vectors

Adding vectors is similar to adding scalars. The individual components of the respective vectors are added to get the final value:

a + b = (a1 \( \hat i\) + b1 \( \hat j\) + c1 \( \hat k\)) + (a2 \( \hat i\) + b2 \( \hat j\) + c2 \( \hat k\)) = (a1, b1, c1) + (a2, b2, c2) = (a1 + a2, b1 + b2, c1 + c2) = (a1 + a2) \( \hat i\) + (b1 + b2) \( \hat j\) + (c1 + c2) \( \hat k\)

The addition of vectors is commutative and associative. There are two laws of vector addition:

Triangle Law of Addition of Vectors: The law states that if two sides of a triangle represent the two vectors (both in magnitude and direction) acting simultaneously on a body in the same order, then the third side of the triangle represents the resultant vector.

Parallelogram Law of Addition of Vectors: The law states that if two co-initial vectors acting simultaneously are represented by the two adjacent sides of a parallelogram, then the diagonal of the parallelogram represents the sum of the two vectors, that is, the resultant vector starting from the same initial point.

Subtraction of Vectors

The subtraction of vectors is similar to the addition of vectors. But here only the sign of one of the vectors is changed in direction and added to the other vector.

a - b = (a1 \( \hat i\) + b1 \( \hat j\) + c1 \( \hat k\)) - (a2 \( \hat i\) + b2 \( \hat j\) + c2 \( \hat k\)) = (a1, b1, c1) - (a2, b2, c2) = (a1 - a2, b1 - b2, c1 - c2) = (a1 - a2) \( \hat i\) + (b1 - b2) \( \hat j\) + (c1 - c2) \( \hat k\)

Scalar Multiplication of Vectors

A scalar is a real number that has no direction. When a scalar is multiplied by a vector, we multiply the scalar by each component of the vector. The operation of multiplying a vector by a scalar is called scalar multiplication. When a vector a = (a1, a2, a3) = a1 \( \hat i\) + a2 \( \hat j\) + a3 \( \hat k\) is multiplied by a scalar r, the resultant vector is:

ra = (ra1, ra2, ra3) = (ra1)e1 + (ra2)e2 + (ra3)e3

• If r is negative, then the direction of the resultant vector changes direction by 180 degrees.
• Scalar multiplication is distributive over vector addition, that is, r(a+b) = ra + rb

The multiplication of vectors with any scalar quantity is defined as 'scaling'. Scaling in vectors only alters the magnitude and does not affect the direction. Some properties of scalar multiplication in vectors are given as,

• k(a + b) = ka + kb
• (k + l)a = ka + la
• a·1 = a
• a·0 = 0
• a·(-1) = -a

Multiplication of Vectors

Vectors can be multiplied but their methods of multiplication are slightly different from that of real numbers. There are two different ways to multiply vectors:

Dot Product of Vectors:

The individual components of the two vectors to be multiplied are multiplied and the result is added to get the dot product of two vectors.
a·b = (a1 \( \hat i\) + b1 \( \hat j\) + c1 \( \hat k\)) ·(a2 \( \hat i\) + b2 \( \hat j\) + c2 \( \hat k\)) = (a1, b1, c1)·(a2, b2, c2) = (a1·a2) + (b1·b2) + (c1·c2)
Another way to determine the dot product of two vectors A and B is to determine the product of the magnitudes of the two vectors and the cosine of the angle between them.
\(\vec{A} ·\vec{B}\) = |A||B| cosθ

The resultant of a dot product of two vectors is a scalar value, that is, it has no direction.

Cross Product of vectors:

The vector components are represented in a matrix and a determinant of the matrix represents the result of the cross product of the vectors.

\(\vec{A} \times \vec{B}\) = (b1c2 - c1b2, a1c2 - c1a2, a1b2 - b1a2)

Another way to determine the cross product of two vectors A and B is to determine the product of the magnitudes of the two vectors and the sine of the angle between them.
\(\vec{A} \times \vec{B}\) = |A||B| sinθ \(\hat{n}\)

Different mathematical operations can be applied to vectors such as addition, subtraction, and multiplication. In this section, we will explore the vector formulas for vector addition, subtraction, dot-product, cross-product and angle between the vectors. The list of vectors formulas that we will be studying in detail further is as follow.

• (a1\(\hat i\) + b1 \( \hat j\) + c1 \(\hat k\)) + (a2 \(\hat i\) + b2 \(\hat j\) + c2 \(\hat k\)) = (a1 + a2) \( \hat i\) + (b1 + b2) \(\hat j\) + (c1 + c2) \(\hat k\)
• (a1\(\hat i\) + b1 \( \hat j\) + c1 \(\hat k\)) - (a2 \(\hat i\) + b2 \(\hat j\) + c2 \(\hat k\)) = (a1 - a2) \( \hat i\) + (b1 - b2) \( \hat j\) + (c1 - c2) \(\hat k\)
• (a1\(\hat i\) + b1 \( \hat j\) + c1 \( \hat k\)) . (a2 \(\hat i\) + b2 \( \hat j\) + c2 \(\hat k\)) = (a1·a2) + (b1·b2) + (c1·c2)
• \(\overrightarrow{A} \times \overrightarrow{B}\) = \(\hat i\) (a2b3 - a3b2) - \(\hat j\) (a1b3 - a3b1) + \(\hat k\) (a1b2 - a2b1)
• θ = cos^-1 (a·b/|a||b|)
• \(\vec A . \vec B = \vec B. \vec A \)
• \( \vec A \times \vec B\neq \vec B \times \vec A \)
• \(\hat i .\hat i =\hat j.\hat j = \hat k.\hat k = 1 \)
• \(\hat i .\hat j =\hat j.\hat k = \hat k.\hat i = 0 \)
• \(\hat i \times \hat i =\hat j\times \hat j = \hat k\times \hat k = 0 \)
• \(\hat i \times \hat j = \hat k~;~ \hat j\times \hat k = \hat i~;~ \hat k\times \hat i = \hat j \)
• \(\hat j \times \hat i = -\hat k~;~ \hat k \times \hat j = -\hat i~;~ \hat i \times \hat k = -\hat j \)

Some of the important properties in vector algebra are as follows.

• The addition of vectors is commutative and associative.
• The dot product of two vectors is a scalar and lies in the plane of the two vectors.
• The cross product of two vectors is a vector, which is perpendicular to the plane containing these two vectors.

Vectors algebra has numerous uses in the field of Physics and Mathematics. Vector algebra deals with quantities which has both direction and magnitude. There are numerous quantities such as velocity, acceleration, force, which need to be represented as expressions in maths, and can be represented as vectors. Some applications of vector algebra are as follows.

• Vectors play a very crucial role in the study of partial differential equations and in differential geometry.
• Vectors are used in physics and engineering, especially in the areas including use of electromagnetic fields, gravitational fields, and fluid flow.
• Vector algebra is useful to find the component of the force in a particular direction.
• Vector algebra is used to find the interplay of two or more quantities in physics.

• Scalar triple product of vectors is the dot of one vector with the cross product of the other two vectors. If any two vectors in a scalar triple product are equal, then the scalar triple product is zero. If the scalar triple product is equal to zero, then the three vectors a, b, and c are said to be coplanar.

  Also, a·(b × c) = b·(c × a) = c·(a × b)

Related Topics on Vectors:

Please check the following links to help us easily learn vectors.

• Vector Subtraction Calculator
• Vector Formulas
• Product of Vectors
• Types of Vectors
• Collinear Vectors
• Addition of Vectors
• Vector Projection Formula
• Dot Product Calculator

What is Vector Algebra?

Vector algebra helps in the representation of geometrical or physical quantities that possess both magnitude and direction in which the object is moving. The magnitude of a vector indicates the length of the vector. It is generally represented by an arrow pointing in the direction of the vector. A vector a is denoted as a1 \( \hat i\) + b1 \( \hat j\) + c1 \( \hat k\), where a1, b1, c1 are its components.

What are the Uses of Vector Algebra In The Physical World?

The physical quantities of displacement, velocity, position, force, torque, are all represented in a three-dimensional plane using the concept from vector algebra. Further the numerous operations of addition, multiplication of these quantities is possible in vector algebra.

How Do We Solve In Vector Algebra?

The vector algebra problems are solved using some of the standard formulas. There are different formulas that can be used to perform operations on vectors in vector algebra. Some of these are as given below,

• \( \hat A = \frac{\vec A}{|\vec{A}|}\)
• \( \vec A·\vec B = |\vec{A}||\vec{B}|cos\theta\)
• \( \vec A \times \vec B = | \vec{A}||\vec{B}|sin \theta \times \hat n\)
• Projection of Vector\( \vec {A} \ \text{on Vector} \vec{B} = \dfrac{\vec{A}. \vec{B}}{| \vec{B}|}\)

What are the Properties of Vectors in Vector Algebra?

There are several properties of vectors in vector algebra. A few of them are as follows.

• Addition of vectors is commutative and associative, that is, ab = ba and a(bc) = (ab)c
• The additive identity of vectors is the zero vector, that is, a + 0 = a
• The additive inverse of a vector is the negative of the vector, that is, a + (-a) = 0
• The scalar multiplication of vectors is associative. r(ab) = (ra)b

How Are the Applications Of Vector Algebra?

In our daily life, you may think of vector algebra being used to represent the velocity of an aircraft, where both the speed and the direction of movement of the aircraft are to be known. Electromagnetic induction involves an interplay of electric forces and magnetic forces.

When are Two Vectors said to be Perpendicular in Vector Algebra?

Two vectors are perpendicular to each other if they are at an angle of 90º to each other. Also, the dot-product of perpendicular vectors is always zero. The angle between two perpendicular vectors is 90°, and the dot product of perpendicular vectors is equal to zero. a.b = |a|.|b|Cos90° = 0.