Vectors and Vector Algebra

Introduction

Vectors are mathematical entities possessing both magnitude and direction. They are essential for describing physical quantities like displacement, velocity, force, and momentum in mechanics. Unlike scalars which have only magnitude, vectors follow specific rules for addition, multiplication, and transformation that reflect the geometric nature of physical space.

Definition and Representation

A vector is a quantity that has both magnitude and direction. Geometrically, it is represented by a directed line segment (arrow). Algebraically, in 3D space: A = A_x\(\hat{\imath}\) + A_y\(\hat{\jmath}\) + Az\(\hat{k}\) where \(\hat{\imath}\), \(\hat{\jmath}\), \(\hat{k}\) are unit vectors along x, y, z axes. The magnitude is |A| = √(A_x² + A_y² + Az²). A unit vector in the direction of A is \(\hat{A}\) = A/|A|.

\n

Introduction

Vectors are mathematical entities possessing both magnitude and direction. They are essential for describing physical quantities like displacement, velocity, force, and momentum in mechanics. Unlike scalars which have only magnitude, vectors follow specific rules for addition, multiplication, and transformation that reflect the geometric nature of physical space.

Definition and Representation

A vector is a quantity that has both magnitude and direction. Geometrically, it is represented by a directed line segment (arrow). Algebraically, in 3D space: A = A_x\(\hat{\imath}\) + A_y\(\hat{\jmath}\) + Az\(\hat{k}\) where \(\hat{\imath}\), \(\hat{\jmath}\), \(\hat{k}\) are unit vectors along x, y, z axes. The magnitude is |A| = √(A_x² + A_y² + Az²). A unit vector in the direction of A is \(\hat{A}\) = A/|A|.

\n

Vector Addition and Subtraction

Vector addition follows the parallelogram law: place vectors tail-to-tail, complete the parallelogram, and the diagonal gives the sum. Triangle method: place tail of second at head of first; sum runs from tail of first to head of second. Algebraically: A + B = (A_x+B_x)\(\hat{\imath}\) + (A_y+B_y)\(\hat{\jmath}\) + (Az+Bz)\(\hat{k}\). Subtraction: A - B = A + (-B) where -B has same magnitude but opposite direction.

Scalar (Dot) Product

The dot product A·B = |A||B|cosθ produces a scalar quantity representing projection. Properties: commutative (A·B = B·A), distributive over addition. Geometrically: A·B equals |A| times component of B along A. Component form: A·B = A_xB_x + A_yB_y + AzBz. Applications: work W = F·d, power P = F·v, finding angle between vectors cosθ = (A·B)/(|A||B|).

\n

Vector (Cross) Product

The cross product A×B = |A||B|sinθ \(\hat{n}\) produces a vector perpendicular to both A and B (right-hand rule determines direction). Magnitude equals area of parallelogram formed by A and B. Properties: anti-commutative (A×B = -B×A), distributive, not associative. Component form using determinant: A×B = \(\hat{\imath}\)(A_yBz - AzB_y) - \(\hat{\jmath}\)(A_xBz - AzB_x) + \(\hat{k}\)(A_xB_y - A_yB_x). Applications: torque τ = r×F, angular momentum L = r×p.

Triple Products and Vector Identities

Scalar triple product: A·(B×C) gives volume of parallelepiped. Cyclic permutation does not change value: A·(B×C) = B·(C×A) = C·(A×B). Vector triple product: A×(B×C) = B(A·C) - C(A·B) (BAC-CAB rule). These identities are essential for simplifying complex vector expressions in electromagnetic theory and fluid mechanics.

\n