4 Geometry and Trigonometry......................................
Geometry is the oldest of the mathematical sciences. Its age-old theorems and the sharp
logic of its proofs make you think of the words of Andrew Wiles, “Mathematics seems to
have a permanence that nothing else has.’’
This chapter is bound to take you away from the geometry of the ancients, with
figures and pictorial intuition, and bring you to the science of numbers and equations
that geometry has become today. In a dense exposition we have packed vectors and their
applications, analytical geometry in the plane and in space, some applications of integral
calculus to geometry, followed by a list of problems with Euclidean flavor but based on
algebraic and combinatorial ideas. Special attention is given to conics and quadrics, for
their study already contains the germs of differential and algebraic geometry.
Four subsections are devoted to geometry’s little sister, trigonometry. We insist on
trigonometric identities, repeated in subsequent sections from different perspectives: Eu-
ler’s formula, trigonometric substitutions, and telescopic summation and multiplication.
Since geometry lies at the foundation of mathematics, its presence could already be
felt in the sections on linear algebra and multivariable calculus. It will resurface again
in the chapter on combinatorics.
4.1 Geometry.....................................................
4.1.1 Vectors.................................................
This section is about vectors in two and three dimensions. Vectors are oriented segments
identified under translation.
There are four operations defined for vectors: scalar multiplicationα−→v, addition
−→v +−→w, dot product−→v ·−→w, and cross-product−→v ×−→w, the last being defined only in
three dimensions. Scalar multiplication dilates or contracts a vector by a scalar. The sum
of two vectors is computed with the parallelogram rule; it is the resultant of the vectors
acting as forces on an object. The dot product of two vectors is a number equal to the