Motion in a Plane

Syllabus Unit: Mechanics

Why it matters

Chapter overview

Two-dimensional motion needs vectors — this topic builds from adding two vectors correctly, to the parabolic path of a projectile launched at an angle, to the always-changing velocity of uniform circular motion, to the relative velocity that governs classic rain-and-boat problems, all of which reduce to the same vector-component toolkit applied in a plane instead of a line.

Physics Syllabus Mechanics 30%–35% weightage

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Resultant Vector

1Vector AFirst vector, magnitude & direction

2Vector BSecond vector, angle θ to A

3Parallelogram LawR = √(A²+B²+2ABcosθ)

4Direction angle αtanα = Bsinθ/(A+Bcosθ)

1Vectors & Vector Addition

Vectors carry both magnitude and direction — NEET's most common vector question is adding two of them correctly with the parallelogram or triangle law.

Learning Objectives

Distinguish scalar and vector quantities and resolve a vector into perpendicular (x, y) components.
Apply the triangle law and parallelogram law of vector addition to find a resultant's magnitude and direction.
Compute the magnitude and direction of the resultant of two vectors using R = √(A²+B²+2AB cosθ) and the tanα formula.
Distinguish vector addition (parallelogram law) from the dot and cross products' distinct outputs (scalar vs vector).

Key Terms

VectorA quantity with both magnitude and direction (e.g. displacement, velocity, force) that obeys the laws of vector addition.

Resultant VectorThe single vector equivalent to two or more vectors added together, found via the parallelogram or triangle law.

ComponentThe projection of a vector along a chosen axis; any vector can be resolved into perpendicular components, e.g. Aₓ = A cosθ, A_y = A sinθ.

Unit VectorA dimensionless vector of magnitude 1 (î, ĵ, k̂) used to indicate pure direction along the x, y, z axes.

In two dimensions, quantities like displacement, velocity and force need both a size and a direction to be fully specified — that's exactly what a vector encodes, and combining two of them correctly is the foundation for everything else in this topic.

Resolving a vector into components

Any vector A at angle θ to the x-axis can be written as A = Aₓî + A_yĵ, with Aₓ = A cosθ and A_y = A sinθ; magnitude A = √(Aₓ²+A_y²).

Adding two vectors

Triangle law — place the tail of the second vector at the head of the first; the resultant is drawn from the tail of the first to the head of the second.
Parallelogram law — place both vectors tail-to-tail; the resultant is the diagonal of the parallelogram they form.

R = √(A² + B² + 2AB cosθ) ; tanα = (B sinθ)/(A + B cosθ)

Key PointSpecial cases — θ=0° gives R=A+B (maximum); θ=180° gives R=|A−B| (minimum); θ=90° gives R=√(A²+B²).

Products of vectors

Dot (scalar) product — A·B = AB cosθ, gives a scalar, used for work = F·d.
Cross (vector) product — A×B = AB sinθ n̂, gives a vector perpendicular to both, used for torque and angular momentum.

Common MistakeVector addition is NOT simple arithmetic addition of magnitudes unless the vectors are parallel — always account for the angle between them.