Vectors Full Chapter Class 11 Physics | NCERT Physics Class 11 Chapter 4 (Part 3)

Vectors: Addition, Resolution, and Motion in a Plane

Physical quantities are classified as either scalars or vectors. Scalar quantities, such as mass, temperature, speed, and energy, have only magnitude. Vector quantities, such as displacement, velocity, acceleration, and force, have both magnitude and direction. Understanding vectors is essential because most physical quantities in mechanics are vectors, and their proper handling is critical for solving problems in physics. This chapter in the CBSE Class 11 Physics syllabus introduces vector algebra, vector addition, resolution of vectors, and applies these concepts to analyse motion in two dimensions (projectile motion and circular motion).

A vector is represented graphically as a directed line segment — an arrow whose length is proportional to the magnitude of the vector and whose direction indicates the vector's direction. Vectors are denoted by bold letters (A) or letters with an arrow on top. The magnitude of vector A is written as |A| or simply A. Equal vectors have the same magnitude and direction, regardless of their position. A unit vector (denoted as â) has a magnitude of 1 and points in the direction of the original vector: â = A/|A|. The three mutually perpendicular unit vectors along the x, y, and z axes are î, ĵ, and k̂ respectively. Vector addition follows the triangle law (placing the tail of one vector at the head of the other) or the parallelogram law (placing both tails at the same point and completing the parallelogram). The resultant R of two vectors A and B is given by: R = √(A² + B² + 2AB cos θ), where θ is the angle between them. Vector subtraction is treated as adding the negative: A − B = A + (−B). Resolution of a vector into components is the reverse process: any vector A making an angle θ with the x-axis can be resolved as Ax = A cos θ and Ay = A sin θ.

Projectile motion is a direct application of vector resolution. When an object is launched with an initial velocity u at an angle θ to the horizontal, the motion can be separated into horizontal (x-direction) and vertical (y-direction) components. The horizontal velocity remains constant (ux = u cos θ, ax = 0), while the vertical velocity changes due to gravity (uy = u sin θ, ay = −g). The equations of motion are applied separately to each direction. The maximum height reached is H = u²sin²θ/(2g), the horizontal range is R = u²sin 2θ/g, and the time of flight is T = 2u sin θ/g. The range is maximum when θ = 45° (since sin 90° = 1), and complementary angles (θ and 90° − θ) give the same range. Uniform circular motion occurs when an object moves in a circle at constant speed — the speed is constant but the velocity continuously changes direction. The acceleration, called centripetal acceleration (ac = v²/r = ω²r), is always directed toward the centre of the circle. The time period T = 2πr/v and frequency f = 1/T. Although the speed is constant, the object is accelerating because the direction of velocity changes — this is a key conceptual point that students often misunderstand. The number of revolutions per second is related to angular velocity by ω = 2πf.

• Scalars have magnitude only; vectors have both magnitude and direction. Represent vectors as directed line segments.
• Parallelogram law: R = √(A² + B² + 2AB cos θ); resolve vectors as Ax = A cos θ, Ay = A sin θ.
• Projectile motion: separate into horizontal (constant velocity) and vertical (acceleration due to gravity) components.
• Range R = u²sin 2θ/g is maximum at θ = 45°; complementary angles give equal ranges.
• Uniform circular motion has constant speed but changing velocity; centripetal acceleration ac = v²/r always points toward the centre.