Mathematics of Physics and Engineering

Inner Product 9

Fig. 1.1.5 Pythagorean Theorem

EXERCISE I.I.6."^4 Prove that the diagonals of a parallelogram intersect at

their midpoints. Hint: let the vectors u and v form the parallelogram and let

r be the position vector of the point of intersection of the diagonals. Argue that

r = u + s(v — w) = t(u + v) and deduce that s = t = 1/2.

1.2 Vector Operations

1.2.1 Inner Product

Euclidean geometry and trigonometry deal with lengths of line segments
and angles formed by intersecting lines. In abstract vector analysis, lengths
of vectors and angles between vectors are defined using the axiomatically
introduced notions of norm and inner product.
In M^3 , where the notions of angle and length already exist, we use these
notions to define the inner product u • v of two vectors. We denote the
length of vector u by ||u||. A unit vector is a vector with length equal to
one. If u is a non-zero vector, then u/||tt|| is the unit vector with the same
direction as u; this unit vector is often denoted by u. More generally, a hat
~ on top of a vector means that the vector has unit length. With the dot •
denoting the inner product of two vectors, we will sometimes write a.b to
denote the product of two real numbers a, b.

Definition 1.2 Let u and v be vectors in M^3. The inner product of u

and v, denoted u • v, is defined by

u • v = ||M||.||U|| cos#, (1.2.1)