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TWO-DIMENSIONAL VECTOR GEOMETRY
(b) Express the vectors r = 2u + v and s = 3u - v in terms of z and w.
7. (a) Prove that, if a, b, c, d are numbers such that ad-bc (0, then au + bv, cu +dv
are linearly independent and, hence, form a basis.
(b) Prove that, if au + by, cu + dv are linearly dependent, then ad - bc = 0 [see
part (a)].
(c) Prove that, if ad - bc = 0, then au + by, cu + dv are linearly dependent.
8. Let w be a nonzero vector and let (a, b)e an ordered number pair. Show that a basis
u, v can be chosen for which w - (a, b) in this basis, for each of the following cases:
(a) a #7 0, b 1 0 (Hint. Choose u so that u, w are linearly independent, then show
that v can be chosen so that w = au + by.)
(b) a 0, b = 0
1-9 ANGLE BETWEEN VECTORS, ORTHOGONAL BASIS
Let u, v be two nonzero vectors in the plane. The anle between u andv is
defined to be the angle BOA, where u = OA, v = OB the angle will usually
be measured in radians, and the value will always be chosen between 0
and 7. The value will be denoted by \ (u, v). It does not depend on the reference point O (Figure 1-29). For if we choose another reference point 0' and
construct A', B', so that O'A' = u, O'B' - v, then the two angles have their
sides parallel and similarly directed; hence, they are eqal. We do not define
the angle between u and v when either vector is 0.
B B'
V \
v
A X
Figure 1-29 Angle between A A'
vectors. 0 0'
In the triangle OAB of Figure 1-29, the sides are Juj, Iv and lu - v; let
4 = (u, v). The law of cosines gives
lu - vF2 = u2+ v lV2 - 2uv 1 cos (1-90)
Thus
u- 12 + v|12 - u - vi2
cosy = (1-91)
2lullvl
When u, v are linearly dependent, the triangle collapses, but (1-90) and
(1-91) still hold true, as we easily verify (see Problem 7 below).
When T = 7r/2, (1-90) reduces to the Pythagorean theorem:
u - vi2 = u12 + Ivl2 (1-92)
Conversely, when (1-92) holds true, (1-91) gives cos - = 0 and, since
0 < T < N7, we have q) = q r/2.
When p = 7</2, the directed line segments representing u are always perpendicular to the ones that represent v. We therefore say that two nonzero
vectors with w/2 as angle are perpendicular or orthogonal. Clearly, two non
