Vector Algebra: Formulas, Types & Operation

Vector algebra is the branch of algebra used to perform algebraic operations on vector quantities.

• It can be represented by two-dimensional and three-dimensional spaces.
• Physical values can be classified as either scalar or vector quantities.
• The vector quantity has both direction and magnitude, and the scalar quantity only possesses magnitude.
• It is used in mathematics, engineering, and physics to perform addition and multiplication operations on physical quantities.
• Length is a vector's magnitude represented with an arrow that indicates the direction.
• It is used to represent a finite sequence of fixed-length numbers.
• You can use the concept to solve problems involving electromagnetic induction, displacement, acceleration and work.

Key Terms: Vector Algebra, Vector, Dot Product, Angle Between Two Vectors, Components of Vectors, Magnitude of Vectors, Vector Algebra Operations

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What is Vector Algebra?

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In mathematics, a vector is a geometric object with magnitude and direction. It has two points: an initial point (A) that indicates its starting location and a terminal point (B) that indicates its ending position. 

• The vector's magnitude is defined as the length of the line connecting points A and B.
• The direction is indicated by the displacement from point A to point B. 
• This is also known as Geometric vectors, Euclidean vectors or Spatial vectors.
• It is derived from the Latin word which means carrier.
• Many algebraic operations, including addition, subtraction, and multiplication, may be carried out using vector algebra.
• Numerous physical quantities, including force and velocity, have both a magnitude and a direction.

Vector Algebra

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Representation of Vector

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Typically, vectors are represented by bold lowercase letters like a or using an arrow above the letter like →a.

• An arrow is used to indicate the beginning and ending points of the physical quantity.
• For instance, vector AB can be represented as \(\overrightarrow{AB}\). 
• It is often represented in the standard form as follows: \(\overrightarrow{A}= a\hat i + b\hat j + c \hat k\) 
• The real values in this case are a, b, and c, while the unit values along the x, y, and z axes are \(\hat i, \hat j,\) and \(\hat k\), respectively.
• A terminal point is referred to as the head, whereas the starting point is known as the tail. 
• In the cartesian coordinate system, these quantities can be denoted by ordered pairs.
• Similarly, vectors in 'n' dimensions can be represented by an 'n' tuple. 
• They are identified with a tuple of components or the scalar coefficients for a set of basis vectors.
• The following are the basis vectors: e₁ = (1,0,0), e₂ = (0,1,0), and e₃ = (0,0,1).

Representation of Vector

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Magnitude of Vector

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One may determine the vector's magnitude by computing the square root of the sum of the squares of its constituent parts. 

• The magnitude formula for vector A is given if its components are (x,y,z).

|A| = √ (x²+y²+z²)

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Components of Vector

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Two vector quantities of the same kind are compared by comparing their respective magnitudes and directions, which are the two features of a vector quantity. 

• A two-dimensional coordinate system may divide a vector into its x and y components. 
• The x-component, V_x, and the y-component, V_y, for a vector, v in the coordinate plane, are shown in the figure below.
• The values of V_x and V_y can be given as V_x = V·cosθ, and V_y= V.sinθ and |V| = √[V_x² + V_y²].

Components of Vector

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Angle between Two Vectors

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The dot product formula may be used to determine the angle between two vectors. Let us take two vectors, a and b, and assume that the angle θ exists between them. 

• The dot product of two vectors can be represented by a·b = |a||b| cosθ.
• The angle between two vectors is used to determine the directions of two vectors.
• The formula to determine the required value θ is as follows:

θ = cos^-1[|a||b|]/(a·b)]

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Types of Vectors - Vector Algebra

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There are various types of vector algebra which are as follows:

Zero Vectors

Zero vectors are defined as vectors with 0 magnitude. It is represented by the symbol \(\overrightarrow{0}\) = (0,0,0). 

• There are no directions and zero magnitudes in the zero vector. 
• Another name for it is the vectors' additive identity.

Unit Vector

Unit vectors are vectors whose magnitude is equal to 1. It is identified by the symbol â. Another name for it is the vector's multiplicative identity. It is often used to indicate a vector's direction.

Position Vector

In three-dimensional space, position vectors are utilized to specify the vectors' location and direction of motion. They can have their magnitude and direction altered in relation to other bodies. A position vector is also known as a location vector.

Equal Vectors

If the equivalent components of two or more vectors are identical, then they are said to be equal. This type of vector has the same direction and magnitude. Their size and direction must be equal, even though their starting and ending positions may differ.

Negative Vectors

Negative vectors refer to two vectors having the same magnitude but different directions. If vectors A and B have the same magnitude but different directions, Vector A is said to be the negative of vector B, or vice versa.

Parallel Vectors

If two or more vectors have the same direction but differ in magnitude, they are referred to as parallel vectors. There is a zero-degree difference in the angles of these quantities. Antiparallel vectors are those whose angles of direction differ by 180 degrees; in other words, they have opposing orientations.

Orthogonal Vectors

Two vectors are said to be orthogonal if they have an angle of 90 degrees with another vector. The dot product of orthogonal vectors can be represented by a·b = |a|·|b|cos 90° = 0.

Co-Original Vectors

Co-initial vectors are vectors that begin at the same location.

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Vector Algebra Operations

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Like normal algebra, we can perform similar algebraic operations on addition, subtraction, and multiplication on vector quantities. Additionally, there are two distinct methods for multiplying vectors: the dot product and the cross product. 

Addition of Vectors

The addition of vectors involves the addition of individual components of the respective vectors to get the required value:

\(a + b = (a_1\hat i + b_1 \hat j + c_1\hat k) + (a_2\hat i + b_2 \hat j + c_2 \hat k)= (a_1, b_1, c_1) + (a_2, b_2, c_2) = (a_1 + a_2, b_1 + b_2, c_1 + c_2) = (a_1 + a_2) \hat i + (b_1 + b_2) \hat i + (c_1 + c_2) \hat i\)

Two key rules that the vector addition adheres to are;

• Commutative Law: A + B = B + A
• Associative Law: A + (B + C) = (A + B) + C

Addition of Vectors

Triangle Law of Vector Addition

According to the triangle law of vector addition, if two vectors represent two sides of the triangle, then the resultant vector is represented by the third side of the triangle.

Triangle Law of Vector Addition

Parallelogram Law of Addition of Vectors

According to the parallelogram law of addition of vectors, if two co-initial vectors are represented by the two adjacent sides of a parallelogram, then the diagonal of the parallelogram is represented by the sum of the two vectors.

Parallelogram Law of Addition of Vectors

Subtraction of Vectors

The subtraction of vectors involves the subtraction of individual components of the respective vectors to get the required value:

\(a - b = (a_1\hat i - b_1 \hat j - c_1\hat k) + (a_2 \hat i - b_2 \hat j = c_2 \hat k) = (a_1, b_1, c_1) - (a_2, b_2, c_2) = (a_1 - a_2, b_1 - b_2, c_1 - c_2) = (a_1 + a_2) \hat i - (b_1 + b_2) \hat i - (c_1 + c_2) \hat i \)

Multiplication of Vectors

Although vectors may be multiplied, the processes involved in doing so differ somewhat from those of real numbers. There exist two distinct methods for multiplying vectors:

Dot Product

A dot product, also known as a scalar product, is represented using a dot(.) between two vectors. 

• In this case, a single integer is obtained by multiplying two coordinate vectors of identical length. 
• In other words, the product of two vectors' scalar products is either a number or a scalar quantity. 
• Assuming that P and Q are two vectors, the dot product of the two vectors is as follows:
• When both P and Q point in the same direction, that is, when θ = 0°, then P.Q = |P| |Q| cos θ
• If θ = 90° indicates that P and Q are orthogonal, then P.Q. equals |P| |Q|
• When two vectors are presented in vector algebra as: P = [P₁,P₂,P₃,P₄,….,P_n] and Q = [Q₁,Q₂,Q₃,Q₄,….,Q_n].
• They then receive their dot product from: P.Q = P₁Q₁+P₂Q₂+P₃Q₃+……….P_nQ_n

\(A = a_1i+a_2j+a_3k\)

\(B = b_1i+b_2j+b_3k\)

\(A \times B = \begin{matrix} i & j & k \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{matrix}\)

Cross Product

The multiplication symbol (×) between two vectors indicates the cross product between two vectors. It is used to define a three-dimensional coordinate system. If A and B are two vectors, then the resultant of the cross product (A x B) is perpendicular to both vectors and normal to the plane that contains vectors. 

A × B = sin θ |A| |B|

Cross Product

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Vector Algebra Formulas

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In vector algebra, we apply different formulae to solve different kinds of complicated problems. These formulae are particularly useful for comprehending and resolving issues with vector algebra. The following are key formulae in vector algebra:

• \((a_1 \hat i + b_1 \hat j + c_1 \hat k) + (a_2 \hat i + b_2 \hat j + c_2 \hat k) = (a_1 + a_2) \hat i + (b_1 + b_2) \hat j + (c_1 + c_2)\hat k\)
• \((a_1\hat i + b_1 \hat j + c_1 \hat k) - (a_2 \hat i + b_2 \hat j + c_2 \hat k) = (a_1 - a_2) \hat i + (b_1 - b_2) \hat j + (c_1 - c_2) \hat k\)
• \((a_1 \hat i + b_1 \hat j + c_1 \hat k) . (a_2 \hat i + b_2 \hat j + c_2 \hat k) = (a_1·a_2) + (b_1·b_2) + (c_1·c_2)\)
• \(\overrightarrow A× \overrightarrow B = \hat i (a_2b_3 - a_3b_2) - \hat j (a_1b_3 - a_3b_1) + \hat k (a_1b_2 - a_2b_1)\)
• \(θ = cos^{-1} \bigg(\frac{a·b}{|a||b|}\bigg)\)
• \(\overrightarrow A.\overrightarrow B=\overrightarrow B.\overrightarrow A\)
• \(\overrightarrow {A}× \overrightarrow B ≠ \overrightarrow B× \overrightarrow A\)
• \(\hat i.\hat i=\hat j.\hat j=\hat k.\hat k=1\)
• \(\hat i.\hat j=\hat j.\hat k=\hat k.\hat i=0\)
• \(\hat i×\hat i=\hat j×\hat j=\hat k×\hat k=0\)
• \(\hat i×\hat j=\hat k ; \hat j×\hat k=\hat i ; \hat k×\hat i=\hat j \)
• \(\hat j×\hat i=−\hat k ; \hat k×\hat j=−\hat i ; \hat i×\hat k=−\hat j\)

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Applications of Vector Algebra

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There are many applications for vector algebra in mathematics and physics, which are as follows:

• In differential geometry and the study of partial differential equations, vectors are extremely important.
• They are particularly useful in the study of gravitational fields, electromagnetic fields, and fluid movement.
• The formula is used to determine the component of the force acting in a specific direction.
• In physics, vector algebra is used to determine how two or more quantities interact with each other.

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Things to Remember

• Vector algebra is an important topic in algebra that expresses both magnitude and direction.
• It is represented by an arrow whose length is directly proportional to the quantity's magnitude and whose direction matches that of the quantity.
• The concept is used to find equipotential surfaces.
• In three dimensions, vector operations determine the equations used to represent lines, planes, and spheres.
• It is used by air traffic controllers for tracking planes.

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