Vector Algebra

Vector algebra is a branch of mathematics that deals with vectors—quantities that have both magnitude and direction. A vector is represented as an arrow—the length of the arrow shows the magnitude and the direction shows the orientation. Vectors are essential for modeling real-world quantities such as velocity, force, and acceleration.  
 

Quantities are crucial in physics and mathematics for explaining the world in which we live. Some quantities, called scalars, have only magnitude. Others, like vectors, have both magnitude and direction. An example of vector quantity is force. 

Scalar Quantity

• A scalar quantity has only magnitude, meaning it has a numerical value with units but no direction. 
• Since scalar quantities are just numbers with units, like 5 cm or 20 kg, we can perform basic operations with them just like regular numbers.  
• A scalar quantity is described by a single value with units, and it depends only on magnitude, not direction.
• Examples: temperature, mass, distance, speed, time, volume, energy.

Vector Quantities

• Vector quantities have both magnitude and direction, and they are typically represented by arrows.
• We need to know vector algebra to work with vectors. Vector algebra includes important rules and operations like vector addition, as well as dot and cross products.
• Vectors are defined in one, two, or three dimensions, depending on the application.
• Examples: force, momentum, electric field, velocity, displacement, acceleration, etc.
   

Zero Vector

Definition: The zero vector has a magnitude of zero with no specific direction.
Notation: 0
Example: 0 = 0i  + 0j + 0k

Unit Vector 

Definition: We use this to indicate direction, and a unit vector has a magnitude of 1.
Notation: A
Example: For A = 3i +4j, A = 32 + 42 = 25 = 5

Position Vector 

Definition: A position vector shows the location of a point in space relative to the origin. It shows the arrow pointing from the origin to the point.
Notation: r
Example: The position vector of a point P(x, y, z) is r = xi  + yj + zk

Co-initial Vectors

Definition: They have the same initial point, even if they point in different directions or have different magnitudes. 
Example: If Vector A and Vector B both start from the origin (0, 0) but point in different directions, then they’re co-initial vectors. 

Collinear Vector

Definition: These lie near the same line or are parallel to each other.
Example: A = 2i + 3j and B = 4i + 6j are collinear.

Equal Vector 

Definition: These have the same magnitude and directions at any point of their initial points.
Example: A = 3i  + 4j and B = 3i  + 4j

Negative of a Vector

Definition: This has the same magnitude as a given vector but points in  the opposite direction.
Example: If A = 5i  + 2j, then the negative of vector A is:
-A = -5i  - 2j

Parallel Vectors 

Definition: Parallel vectors have the same or exactly opposite direction. They may have different magnitudes, but their directions are aligned. 
Example: Vector A and KA, where k is a scalar, are parallel.

Orthogonal Vectors
 
Definition: These vectors are perpendicular to each other, meaning they form a 90o degree angle when they intersect. 
Example: If A = i  + 2j and B = -2i  + j, then AB = 0.

 Coplanar Vector

Definition: This lies in the same plane.
Example: Vectors A = 2i  + 3j and B = -i  + 4j are coplanar with any vector in the xy-plane.
 

Vectors are written in bold (a) or else with an arrow (a), or by their tail and head, like BC. Their standard form is A = ai  + bj + ck with i , j,and k as unit basis factors. Magnitude of a vector 
A=(x, y, z)
|A| = x2+y2+z2  

Components: A vector is separated into parts along each axis—x, y, and z. These parts or components show how much of the vector acts in each direction.

V=(Vx,Vy,Vz) Where Vx,Vy,Vz are the components of the vectors with the x-axis, y-axis, and z-axis. This comes from trigonometry: 

Vx=|V| cos 
Vy=|V| cos 
Vz=|V| cos 

Magnitude of a vector: The Magnitude of a vector is its length and norm, which defines how long the vector is and how far it extends from its starting point to its endpoint. For vector v with components, V=(Vx,Vy) it's calculated like;
|V|=Vx2+Vy2

Direction of a vector: Directions tell us where the vector points, and it can be described using angles from the coordinate axes or by using unit vector components. For a vecto

Vector algebra includes several basic operations like addition, scalar multiplication, cross product, and dot product. These operations help us understand and solve problems in fields like engineering, physics, and computer science.

Vector addition: To add two vectors, such as AB and CD, we can use methods like the triangle law or parallelogram law. 

• Triangle law: Start at A and draw the first vector to point B. Now, from point B, draw the second vector to point C. The sum of these two vectors goes directly from A to C—this line completes the triangle and represents the total vector.
• Parallelogram law of addition of vectors: Draw both vectors from the same source, if they are the two sides of a Parallelogram, now draw the diagonal which starts from their given corner. This diagonal will be exactly the sum of two vectors.

Vector subtraction: To calculate A - B, reverse B in the opposite direction (-B), and then add it to A: 
A - B = A + (-B)
The vector -B has the same length as B but points in the opposite direction. Now, after subtracting them, the vector a  b is shown as the addition of  a and -b. We write this as:
 
a  -b= a+ (-b)

Scalar Product: There is a and b non-zero vectors at an angle . Now it is denoted as a b and will be defined as a b=|a||b|cos ; 0 .

Vector Addition 
          A+B = (Ax+Bx) i +(Ay+By)j +(Az+Bz)k
          Add every pair of matching components of vectors A and B.

Vector Subtraction
            A-B = (Ax-Bx) i +(Ay-By)j +(Az-Bz)k
            Subtract every similar component of vector B and A.

Scalar Multiplication
           kA = (kAx) i +(kAy)j +(kAz)k
           Multiply every component of the vector by the scalar k.

Dot Product
AB=|A||B| cos 

If A and B are perpendicular, then A B will be 0. 
If A and B are parallel, then their dot product is not zero unless either A or B is a zero vector. 
 

The concept of vector algebra is used in many fields such as structural engineering, aerospace engineering, and sports science. Some of the applications are mentioned here:

• Structural Engineering: Civil and structural engineers use vector algebra to analyze and ensure stability of structures under pressure exerted by different forces. For example, on a bridge, engineers resolve applied loads into x- and y-components to sum them to calculate the resultant force. They use this information to design the support structures accordingly.

• Aerospace and Navigation System: Pilots, aerospace engineers, and navigation system designers compute the exact trajectory, speed, and position when many motion vectors like velocity, and wind to ensure accuracy in navigation. 

• Sports Science and Ballistics: Coaches, sports analysts and athletes use this to optimize trajectory, distance, and angles of projectiles by decomposing motions into vector components. For example, a batsman or bowler in cricket analyzes launch velocity and angle using decomposition to predict ball flight path and landing.

• Computer Graphics and video Games: Game developers, graphics programmers use this to control object motion, collision detection, shading, camera direction and more using vector maths. For example, in a game, position and movement of characters are vector-based. Ray casting uses direction vectors to check bullet hits, dot products to check if an object lies within a camera's forward direction.

• Robotics and path planning: Roboticists and automation engineers use this to guide robot motion, avoid obstacles and complete necessary movement vectors from current to target positions. For example, a robot calculates the displacement vector target position - current position and adjusts it with additional vectors to follow as a safe p