A vector–valued function for represents the orientation of a rigid body in space and…

MATHEMATICS · Multi-part · AP · Calculus BC · Parametric, Polar, and Vector-Valued Functions · Parametric, Polar, and Vector-Valued Functions

Problem

A vector–valued function

\mathbf u(t)

for

t\ge 0

represents the orientation of a rigid body in space and satisfies the identity

\mathbf u(t)\cdot\mathbf u(t)=1

for all

t

. (a) Differentiate the identity

\mathbf u(t)\cdot\mathbf u(t)=1

with respect to

t

and write the resulting relationship between

\mathbf u(t)

and

\mathbf u'(t)

. Explain, in words, what this relationship implies about the geometric connection between

\mathbf u(t)

and

\mathbf u'(t)

at every time

t

. (b) Let

\mathbf u(t)=\bigl(\cos(2t),\,\sin(2t),\,0\bigr)

. Verify explicitly that

\mathbf u(t)\cdot\mathbf u(t)=1

and use your result from part (a) to confirm that

\mathbf u'(t)

is perpendicular to

\mathbf u(t)

for all

t

. Show all steps in your reasoning. (c) Interpret the result of part (b) in the context of motion along a circle of radius

1

. In particular, describe how the direction of

\mathbf u'(t)

relates to the motion of a point moving with position vector

\mathbf u(t)

. Justify your response with reference to the relationship you found in part (a).