Vectors and Scalars
Distinction between scalar and vector quantities, geometric representation, vector addition (triangle law, parallelogram law), subtraction, and resolution into mutually perpendicular rectangular components.
Marking a lesson complete represents finishing your study of the topic; it does not denote tested mastery.
Learning Objectives
- Distinguish rigorously between scalars and vectors based on physical transformation laws.
- Resolve vectors along mutually perpendicular axes: Ax = A cos θ, Ay = A sin θ.
- Determine the resultant magnitude and direction of coplanar concurrent vectors analytically.
Core Theory & Concepts
Physical quantities are categorized into scalars and vectors. A scalar has only magnitude and unit (e.g. mass, time, temperature). A vector has magnitude, unit, and an essential spatial direction, obeying the geometric laws of vector addition (e.g. displacement, velocity, force).
Why Current is a Scalar
Although electric current has a directional flow along a wire, it does NOT obey the parallelogram law of vector addition. Current at a junction combines purely algebraically (I = I₁ + I₂) according to Kirchhoff’s First Law, regardless of conductor angles.
Rectangular Resolution
Any vector A making an angle θ with the positive x-axis resolves into perpendicular components: Ax = A cos θ (horizontal) and Ay = A sin θ (vertical). Magnitude is restored via Pythagoras: A = √(Ax² + Ay²).
Parallelogram Law of Addition
For two vectors A and B with angle θ between them, the resultant magnitude is: R = √(A² + B² + 2AB cos θ). If perpendicular (θ = 90°), R = √(A² + B²).
Vector Subtraction & Relative Velocity
Subtracting vector B from vector A is defined as adding the negative vector: D = A - B = A + (-B). The negative vector -B has equal magnitude but opposite direction (rotated 180°). In kinematics, relative velocity of body A with respect to body B is given by vector subtraction: v_AB = v_A - v_B.
Polygon Law of Vectors & Equilibrium
If any number of coplanar vectors are represented in magnitude and direction by the sides of an open polygon taken in order (head-to-tail), their resultant is represented in magnitude and direction by the closing side drawn from the tail of the first vector to the head of the last vector. If the vectors form a closed polygon, the resultant is zero (ΣF = 0), representing a state of static or dynamic equilibrium.
Common Misconceptions
Any physical quantity that has a direction is automatically a vector.
Syllabus Topic Outline
- 1Scalars versus vectors definition and examples
- 2Parallelogram Law of vector addition: R = √(A² + B² + 2AB cos θ)
- 3Triangle Law and Polygon Law of concurrent vectors
- 4Resolution into rectangular components Ax and Ay
- 5Resultant calculation: Rx = ΣAx, Ry = ΣAy, R = √(Rx² + Ry²)
Interactive 2D Vector Workspace
Geometric representation, rectangular component resolution, addition, subtraction & polygon law