Vector Algebra

Detailed Notes on Vector Algebra

10.1 Introduction

• Scalars vs. Vectors: Scalars have only magnitude, while vectors have both magnitude and direction. Examples include:
  • Scalars: length, mass, time, speed, area
  • Vectors: displacement, velocity, acceleration, force

10.2 Some Basic Concepts

• Directed Line Segments: A directed line segment consists of a line with an initial point (A) and terminal point (B).
• Vector Definition: A quantity with both magnitude and direction is called a vector, denoted as AB.*
• Position Vector: A vector from the origin O to the point P with coordinates (x, y, z) is the position vector, denoted as: [ |\vec{OP}| = \sqrt{x^2 + y^2 + z^2} ]
• Direction Cosines: The angles a vector forms with the coordinate axes, with cosines termed direction cosines (l, m, n). If the direction angles are α, β, γ, then:
  • l = cosα, m = cosβ, n = cosγ
  • The relationship holds: [ l^2 + m^2 + n^2 = 1 ]

10.3 Types of Vectors

• Zero Vector: A vector where the initial and terminal points coincide. It cannot have a specific direction.
• Unit Vector: A vector with magnitude equal to 1.
• Coinitial Vectors: Vectors that share the same initial point.
• Collinear Vectors: Vectors that lie along the same line (either same or opposite directions).
• Equal Vectors: Vectors that have the same magnitude and direction, regardless of their position.
• Negative Vector: A vector that has the same magnitude as a given vector but points in the opposite direction.

10.4 Addition of Vectors

• Triangle Law: To add vectors, arrange them in a sequence, for example, A to B to C, and the resultant vector is from A to C.
• Parallelogram Law: The sum of two vectors can be represented by the diagonal of the parallelogram formed by the two vectors.
• Properties of Vector Addition:
  • Commutative: [ \vec{a} + \vec{b} = \vec{b} + \vec{a} ]
  • Associative: [ \vec{a} + (\vec{b} + \vec{c}) = (\vec{a} + \vec{b}) + \vec{c} ]
  • Additive Identity: There exists a zero vector such that [ \vec{a} + \vec{0} = \vec{a} ]

10.5 Multiplication of a Vector by a Scalar

• Scalar Multiplication: When a vector is multiplied by a scalar λ, the direction may change based on the sign of λ, while the magnitude is multiplied by |λ|. The unit vector in the direction of vector a is given by: [ \hat{a} = \frac{\vec{a}}{|\vec{a}|} ]
• Components of a Vector: In a 3D space, [ \vec{a} = x\hat{i} + y\hat{j} + z\hat{k} ]
  • Where (x,y,z) are the scalar components of the vector along the x, y, z axes respectively.

10.5.1 and 10.5.2 Additional Concepts

• Direction Ratios: Proportional to direction cosines and are related to the components of the vector.
• Vector Joining Two Points: The vector from point P1 to point P2 is given by [ \vec{P_2P_1} = (x_2 - x_1)\hat{i} + (y_2 - y_1)\hat{j} + (z_2 - z_1)\hat{k} ]
• Section Formula: Calculates the position vector of a point dividing a line segment in a given ratio, both internally and externally.

10.6 Product of Two Vectors

• Scalar Product: Given by ( a\cdot b = |a||b|cosθ ).
  • Indicates the angle between vectors. Perpendicular vectors yield a product of zero.
• Vector Product: Defined as ( a \times b = |a||b|sinθ , \hat{n} ), where n is a unit vector perpendicular to both a and b.
• Area Representations: Area of a triangle or parallelogram formed by vectors can be derived from the magnitude of their products.

Key Points

1. Scalars vs. Vectors: Scalars are magnitude only; vectors have both magnitude and direction.
2. Vectors can be represented in component form as ( \vec{a} = x \hat{i} + y \hat{j} + z \hat{k} ).
3. Direction Cosines: Represent direction of vectors relative to axes ( ( l, m, n ) ).
4. Addition of Vectors: Can be achieved through Triangle Law or Parallelogram Law.
5. Unit vectors indicate direction without regard to magnitude.
6. Multiplying a vector by a scalar changes its magnitude and can alter its direction.
7. Dot product yields a scalar; cross product yields a vector.
8. Zero vector has no direction; unit vector has a magnitude of one.
9. The resultant vector represents the sum of two or more vectors either graphically or algebraically.
10. Section formula provides a method to find a point along a line segment based on a ratio.

Conclusion

Vector algebra underpins various fields in physics, engineering, and mathematics, emphasizing the understanding of both theoretical properties and practical applications.

1. Scalars vs Vectors: Scalars have only magnitude; vectors have both magnitude and direction.
2. Addition of vectors can be done using the Triangle Law or Parallelogram Law.
3. Position Vector of a point gives the vector from the origin to that point.
4. Direction Cosines are the cosines of angles formed by the vector with the coordinate axes.
5. Unit Vectors have a magnitude of 1 and indicate direction alone.
6. The Dot Product (scalar product) and Cross Product (vector product) are important operations on vectors.
7. The Zero Vector is a unique vector with no magnitude or direction.
8. The Scalar Multiplication alters the magnitude of a vector, possibly changing its direction.
9. Direction Ratios relate to the direction cosines and components of a vector.
10. The Section Formula allows finding points dividing line segments in specific ratios.