Chapter 10 โ Vector Algebra
Displacements, velocities and forces have a size and a direction. This chapter turns such arrows into objects you can add, stretch and multiply โ and uses them to find angles, projections, work, areas and directions in three-dimensional space.
A ferry leaves a ghat on the Brahmaputra and heads straight for the far bank, but the river is flowing fast; a pilot lining up on a runway feels a crosswind pushing the aircraft sideways; an ISRO engineer pointing a satellite antenna needs the direction perpendicular to a solar panel. In each case the important quantities are arrows โ a velocity, a wind, a direction in space โ and the questions are about combining them: where does the ferry actually go? How much of the wind blows along the runway and how much across it? Which way does the panel face, and how much sunlight can it catch?
This chapter builds the algebra of such arrows: vectors. You will describe a vector in space by three components along $\hat i,\hat j,\hat k$, its length by a square root and its direction by direction cosines. You will add vectors by the triangle law, stretch them by scalars, write the vector joining two points and the point dividing a segment in a given ratio. Then come two different ways of multiplying vectors: the scalar (dot) product $\vec a\cdot\vec b=|\vec a|\,|\vec b|\cos\theta$, a number that measures angles, projections and the work done by a force, and the vector (cross) product $\vec a\times\vec b$, a vector perpendicular to both whose length is the area of the parallelogram they span. Chapter 11 โ Three Dimensional Geometry then uses all of this to describe lines in space.
- tell scalars from vectors, recognise zero, unit, equal, negative, collinear and coinitial vectors, and write any vector in component form $a_1\hat i+a_2\hat j+a_3\hat k$;
- find the magnitude, direction cosines and direction ratios of a vector, use $l^2+m^2+n^2=1$, and find unit vectors and vectors of a given length in a given direction;
- add and subtract vectors by the triangle and parallelogram laws and in components, find the vector joining two points, test points for collinearity, and use the section formula (internal and external) and the centroid formula;
- compute scalar products, angles between vectors, perpendicularity conditions, projections and the work done by a constant force, and use identities such as $|\vec a+\vec b|^2=|\vec a|^2+2\,\vec a\cdot\vec b+|\vec b|^2$;
- compute vector products by a determinant, find unit vectors perpendicular to two vectors, and find areas of parallelograms and triangles in space;
- answer the Board's 1-, 2-, 3- and case-study questions on vectors and the JEE Main / JEE Advanced problems built on them.
Sections in this chapter
Vectors, Their Types and Components
Scalars and vectors; position vectors; direction cosines and direction ratios; zero, unit, equal, negative, collinear and coinitial vectors; components along $\hat i,\hat j,\hat k$.
Addition, Scalar Multiples and the Section Formula
The triangle and parallelogram laws; multiplying by a scalar; operations in components; the vector joining two points; collinearity; internal and external division.
The Scalar (Dot) Product and Projections
$\vec a\cdot\vec b$ as $|\vec a|\,|\vec b|\cos\theta$ and as $a_1b_1+a_2b_2+a_3b_3$; angles and perpendicularity; projections; work done; identities and inequalities.
The Vector (Cross) Product
The right-hand rule; properties; the determinant formula; unit vectors perpendicular to two vectors; areas of parallelograms and triangles; Lagrange's identity.