Chapter 3: Motion in a Plane

3.1 Introduction

In this chapter, we’ll explore how to describe motion in two dimensions using vectors. Unlike motion along a straight line (where direction is just + or -), motion in a plane requires vectors to represent quantities like displacement, velocity, and acceleration. We’ll also study projectile motion and circular motion as special cases.

3.2 Scalars and Vectors

Scalar Quantities

  • Have only magnitude (size) and no direction.
  • Examples: Distance (5 m), mass (2 kg), temperature (25°C), time (10 s).
  • Follow normal math rules: You can add, subtract, multiply, and divide them like regular numbers.
  • Example: If a rectangle has sides 1.0 m and 0.5 m, its perimeter is 1.0 + 0.5 + 1.0 + 0.5 = 3.0 m.

Vector Quantities

  • Have both magnitude and direction.
  • Follow special rules for addition (triangle or parallelogram law).
  • Examples: Displacement (5 m north), velocity (20 m/s east), force (10 N upwards).
  • Represented by bold letters (v) or with an arrow (→v). Magnitude is written as |v| or just v.

Position and Displacement Vectors

  • Position vector (r): Points from the origin (O) to an object’s location (P).
  • Displacement vector (Δr): Represents change in position (from P to P’). It’s the straight-line path between initial and final points, regardless of the actual path taken.
  • Example: If you walk in a zigzag but end up 10 m north, your displacement is still 10 m north.

Equality of Vectors

  • Two vectors (A and B) are equal only if they have the same magnitude and direction.
  • Example: If you move B parallel to itself and it perfectly overlaps A, they’re equal.
  • Note: Vectors with the same magnitude but different directions (e.g., 5 m east vs. 5 m west) are not equal.

3.3 Multiplying Vectors by Numbers

  • Multiplying a vector A by a positive number (λ) scales its magnitude but keeps the direction the same: |λA| = λ|A|.
  • Example: 2A is twice as long as A and points the same way.
  • Multiplying by a negative number flips the direction: -1.5A is 1.5 times as long but points opposite to A.

3.4 Adding Vectors Graphically

  • Triangle Law: To add A + B, place the tail of B at the head of A. The sum R is the vector from the tail of A to the head of B.
  • Parallelogram Law: Draw vectors A and B tail-to-tail. Complete the parallelogram. The diagonal from the tails to the opposite corner is A + B.
  • Order doesn’t matter: A + B = B + A.

Important Concepts for NEET

  1. Scalars vs. Vectors: Know the difference (direction matters for vectors!).
  2. Displacement vs. Path Length: Displacement is the shortest distance between points, regardless of the path taken.
  3. Vector Addition: Master the triangle/parallelogram methods for graphical problems.
  4. Multiplying Vectors: Understand how positive/negative numbers affect magnitude and direction.
  5. Equality of Vectors: Same magnitude + same direction = equal vectors.

Keep practicing drawing vectors—it’ll make projectile and circular motion much easier later!