Calculus: Analytic Geometry and Calculus, with Vectors

2.6 Vector products and changes of coordinates in E3 107

and therefore

(5) x = x' cos 0 - y' sin 0
y = x' sin 0 + y' cos 4).

Remark: The formulas (5) are, perhaps unwisely, called "formulas for rotation
of axes" in E2. Actually, they are used to convert equations involving "old"
coordinates x, y into new (and sometimes simpler) equations involving new
coordinates x', y'.
14 Supposing that u and v are nonzero noncollinear vectors, show that the
vector
(u X V) X U

I(uXv)XuI

is a unit vector which lies in the plane of u and v and is orthogonal (or perpen-
dicular) to U.
15 Cultivate some useful skills by following instructions and paying particu-
lar attention to steps that seem to be worthy of notice. Draw vectors PP2 and
Pp3 and then draw the angle 0 and the unit normal n that appear in the definition
of PP-2 X PP3. Show that the area A of the parallelogram having adjacent sides
on PP2 and PP3 is
A = IPP2I IPP3I sin 0.

Draw another vector PPl and show that the distance d from Pl to the plane of

pp2 and FP3 is

d =

where the sign is so taken that d? 0. Then, depending upon circumstances,

remember or learn that V = 4d, where V is the volume of the parallelepiped

having adjacent edges on PP,, PP2, PP3 if d 0 0 and V = 0 if Pi lies in the plane

of PP2 and PP3. Use this to show that

V = X PP3).

Supposing that P has coordinates x, y, z and that the points Pk have coordinates

Xk, yk, Zk, show that

V = ±[(x, - x)i + (y' - y)j + (zl - z)k]-

i j k

x2-x y2-y z2-z

xa - x y3-y z3-z

and hence that

x, - x yl - y zI - z

V = ± x2-x y2-y z2-z

X3 - X y3-y z3-z

Use this to show that

x y z i x y z^1

Y- + x, - x yi - y zl - z^0

xl yi zI 1

x2-x y2-y z2-z 0 X2 y2 z2^1

x3-x y3-y z3-z 0 X3 ya z3^1