Vector Algebra

Vector algebra is a branch of mathematics that deals with quantities having both magnitude and direction. Vectors are used to represent physical quantities such as force, velocity, and displacement. This chapter introduces the basic concepts, operations, and applications of vectors.

1. Introduction to Vectors

• Scalar: A quantity with only magnitude (e.g., mass, temperature).
• Vector: A quantity with both magnitude and direction (e.g., force, velocity).
• Representation: Vectors are represented by directed line segments or in component form as a = a₁i + a₂j + a₃k.

2. Types of Vectors

• Zero Vector: Magnitude is zero.
• Unit Vector: Magnitude is one.
• Position Vector: Represents the position of a point with respect to the origin.
• Like and Unlike Vectors: Same or different directions.
• Collinear Vectors: Parallel to the same line.
• Coplanar Vectors: Lie in the same plane.

3. Operations on Vectors

• Addition: Triangle law and parallelogram law of vector addition.
• Subtraction: Adding the negative of a vector.
• Multiplication by a Scalar: Changes the magnitude, not the direction (unless the scalar is negative).

4. Components of a Vector

• Any vector in 3D can be written as a = a₁i + a₂j + a₃k.
• Magnitude: |a| = √(a₁² + a₂² + a₃²).

5. Section Formula

• Dividing a line segment in a given ratio using vectors.
• If point P divides AB in the ratio m:n, then OP = (mOB + nOA)/(m+n).

6. Scalar (Dot) Product

• a · b = |a||b|cosθ, where θ is the angle between a and b.
• Result is a scalar.
• Properties: Commutative, distributive over addition.
• Applications: Work done, projection of vectors.

7. Vector (Cross) Product

• a × b = |a||b|sinθ n̂, where n̂ is a unit vector perpendicular to both a and b.
• Result is a vector.
• Properties: Anticommutative, distributive over addition.
• Applications: Area of parallelogram, torque.

8. Applications of Vectors

• Physics: Force, velocity, acceleration.
• Geometry: Finding distances, areas, and angles.
• Engineering: Mechanics, navigation, robotics.

9. Practice Problems

1. Find the magnitude of the vector a = 3i + 4j + 12k.
2. If a = 2i + 3j and b = i - 2j, find a + b and a - b.
3. Calculate the dot product of a = i + 2j + 3k and b = 4i - j + k.
4. Find a unit vector in the direction of a = 6i - 8j.
5. If a = i + j and b = i - j, find the angle between a and b.

10. Summary

• Vectors have both magnitude and direction.
• Basic operations: addition, subtraction, scalar and vector products.
• Applications in physics, geometry, and engineering.