Calculus: Analytic Geometry and Calculus, with Vectors

10.2 Polar curves, tangents, and lengths 541

The vector ul(t) is the unit vector in the direction of the projection of
the vector r(t) upon the xy plane. The vector u2(t) is easily seen to be
another unit vector, and it is orthogonal to both ul(t) and k because

0, 0.

Moreover, as we see by introducing vector products,

uI(t) x u2(t) =

i j k

cos 4,(t) sin q(t)^0

• sin 4.(t) cos 0(t) 0

= k,

so the three vectors ul(t), u2(t), k, in this order, constitute a right-

handed orthonormal system of vectors. Figure 10.27 shows these things

x

Figure 10.27

for the special case in which C, and hence the vector r(t), lies in the xy

plane of the paper and the unit vector k therefore extends vertically

upward from the plane of the paper. Use of the right-hand finger and

thumb rule shows that Figure 10.27 would be wrong if the directionof

the vector u2(t) were reversed, and this shows that our excursion into

E3 helps us to see how things are oriented in the plane.

Thus, (10.261) and (10.262) are remarkably simple and informative

formulas which display the scalar and vector components ofv(t), the

velocity vector or the forward tangent vector, in terms of the three

orthonormal vectors ui(t), u2(t), and k. The orthonormality of these

vectors enables us to use (10.26) and (10.261) toobtain the formulas

(10.271)

(10.272) Io(t)I = [P'(t)]2 + [P(t)q5'(t)]2 + [z'(t)]2

(10.273) P(t)P(t) + z(t)z'(t).

The angle ¢ between the vectors r(t) and v(t) can, when Ir(t)I 0 0 and

Jv(t)I 0 0, be calculated from the basic formula

(10.274) Ir(t)I Iv(t)I cos ¢.