Problems

Find the vector equations for the motion of the given point in the plane. The positions at t = 0 and t = 1 are as shown in the figures.

	A point moving along the parabola y = x2 in such a way that x = 3t.
	A point moving along y = x2 so that xy = t.
	A point moving upward along the line y = 2x so that its distance from the origin at time t is t3.
	A wheel of radius one is turning at the rate of one radian per second. At the same time its center is moving along the x-axis at one unit per second. Find the motion of a point on the circumference of the wheel.
	The point at distance one from the origin in the direction of the point (t, 1).
	The point where the parabola y = x2 intersects the line through the origin which makes an angle t with the x-axis.
	The point halfway between a point P going around the circle x2 + y2 = 1 at one radian per second and a point Q going around the same circle at 3 radians per second.
	A wheel of radius one rolls along the x-axis at one radian per second. Find the motion of a point on the circumference of a concentric axle of radius ½.
	A circle of radius one rolls around the outside of the circle x2 + y2 = 9 at one radian per second. Find the motion of a point on the circumference of the smaller circle.
	Find the motion of the point in Problem 9 if the small circle rolls around the inside of the large circle.
	A string is unwound from a circular reel of radius one at one radian per second. The string is held taut and forms a line tangent to the reel. Find the motion of the end of the string.

In Problems 12-23, find the vector equation for the motion of the given point in space.

12            A point moving so that at time t its position vector has length t² and direction cosines

(⅓, ⅔, ⅔)

13            A point X moving at one radian per second counterclockwise around a horizontal unit circle whose center is at (0, 0, t²) at time t. (At t = 0, X = i.)

14            The point which at time t is at distance one from the origin in the direction of the vector ti + j + t²k.

15            The point at distance one from the point P(1,2,1) in the direction of the vector t²j + (t² - 1)k.

16            The point where the line through the origin in the direction of i + tj + t²k intersects the plane x + 2y + 3z = 1.

17            The point halfway between a point P going around the circle x² + y² = 1 in the (x, y) plane at one radian per second and a point Q going around the circle x² + z² = 1 in the (x, z) plane at 2 radians per second. (At t = 0, P = Q = i. Both motions are counterclockwise.)

18            The point at distance f(t) from the point P(t) in the direction of the vector D(t).

19            The point on the plane x + y + z = 1 which is nearest to the point

cos ti + sin tj + 6k.

20            The point where the rotating plane x cos t + y sin t = 0 intersects the line through (1, 1, 1) and (2, 3, 4).

21             The point on the rotating plane x cos t + y sin t = 0 which is nearest to the point ti + 2tj + 3tk.

22            Find the price vector P(t) for three commodities such that the first has price 1/t, the second has double the price of the first, and the sum of the prices is 4 (t ≥ 1).

23             Find the price vector P(t) of three commodities such that the product of the three prices is one. the first commodity has price 2t, and the third commodity has price t + 1 (t ≥ 1).