Mathematics of Physics and Engineering

Inner Product 13

orthogonal to any vector. This is consistent with (12): taking A — \i — 1
and v = 0, we also find w • 0 = 0 for every w.

EXERCISE 1.2.2.C Prove the law of cosines: a^2 = b^2 + c^2 - 2bccos6, where
a, b, c are the sides of a triangle and 6 is the angle between b and c. Hint:
Let c = ||n||, 6 = ||r 2 ||. Then a^2 = \r 2 - rif = (n - r 2 ) • (n - r 2 ).
We now discuss some APPLICATIONS OF THE INNER PRODUCT. We start
with the EQUATION OF A LINE IN R^2. Choose an origin O and drop the
perpendicular from O to the line L; see Figure 1.2.4.

Fig. 1.2.4 Line in The Plane

Let n be a unit vector lying on this perpendicular. For any point P on
L, the position vector r satisfies

rn = d, (1.2.7)

where \d\ is the distance from O to L; indeed, |r • n\ is the length of the
projection of r on n. In a cartesian coordinate system (x,y), r = xi + yj,
and equation (1.2.7) becomes ax + by = d, where n = ai + bj. More
generally, every equation of the form a\x + a^y — 013, with real numbers
ffli, 0,2,0,3, defines a line in M^2.
Similar arguments produce the EQUATION OF A PLANE IN R^3. Let n be
a unit vector perpendicular to the plane. For any point P in the plane, the
equation (1.2.7) holds again; Figure 1.2.4 represents the view in the plane
spanned by the vectors n and r and containing points O, P. In a cartesian
coordinate system (x,y,z), r = xi+yj+zk,, and equation (1.2.7) becomes
ax + by + cz = d, where n = ai + bj+ck. More generally, every equation of
the form a\x + a,2y + a^z = 04 defines a plane in R^3 with a (not necessarily
unit) normal vector ax i + a-i 3 + az k. For alternative ways to represent
a line and a plane see equations (1.1.5) and (1.1.6) on page 8.