Class 11 Physics — Chapter 301: Mathematical Methods

Key concepts: Mathematical Methods (Class 11 Physics)

This chapter builds the mathematical toolkit that runs through all of Std 11 physics. It separates scalars from vectors, shows how vectors are represented and combined by the triangle, parallelogram and polygon laws, and how they are resolved into rectangular components using unit vectors î, ĵ and k̂. It introduces the scalar (dot) and vector (cross) products, and gives a first taste of calculus — the derivative as a rate of change and integration as area or anti-derivative — closing with relative velocity as a vector application.

Scalars and vectors

A scalar has only magnitude (mass, speed, temperature), while a vector has both magnitude and direction (displacement, velocity, force) and obeys the laws of vector addition.

Representation of a vector

A vector is drawn as an arrow whose length (to a chosen scale) gives its magnitude and whose arrowhead gives its direction; in symbols it is written with a bar or arrow over the letter.

Triangle law of addition

If two vectors are the two sides of a triangle taken in the same order, their resultant is the third side taken in the opposite order, joining the start of the first to the end of the second.

Parallelogram law

If two vectors from a common point form the adjacent sides of a parallelogram, the resultant is the diagonal drawn from that same common point, useful for forces acting together.

Polygon law of addition

To add three or more vectors, place them head to tail in turn; the resultant is the single vector that closes the polygon from the tail of the first to the head of the last.

Subtraction of vectors

Subtracting vector B from A means adding the negative of B, so A − B = A + (−B), where −B has the same magnitude as B but the exactly opposite direction.

Resolution into components

Any vector can be split into perpendicular rectangular components along the axes; for a vector A at angle θ to the x-axis, Ax = A cosθ and Ay = A sinθ recombine to give A.

Unit vectors î, ĵ, k̂

These are mutually perpendicular vectors of magnitude one pointing along the x, y and z axes, so any vector can be written as A = Ax î + Ay ĵ + Az k̂.

Scalar (dot) product

Defined as A·B = AB cosθ, the dot product is a scalar; it is zero for perpendicular vectors and is used to define work done by a force as F·d.

Vector (cross) product

Defined with magnitude |A×B| = AB sinθ, the cross product is a vector perpendicular to both A and B, its direction fixed by the right-hand rule; it is zero for parallel vectors.

Derivative as rate of change

The derivative dy/dx measures the instantaneous rate at which one quantity changes with another; in physics velocity is dx/dt and acceleration is dv/dt.

Integration

Integration is the reverse of differentiation (anti-derivative) and also gives the area under a curve; for example displacement is the integral of velocity over time.

Relative velocity

The velocity of body A relative to body B is the vector difference vAB = vA − vB, so bodies moving in the same direction subtract and those in opposite directions add.