Calculus-Sec-12-1-Solution

12.1      Vector-Valued Functions and Space Curves

by SGLee - HSKim - JHLee

1-2. Find the domain of the vector functions.

1.

.

2.

and .

Thus  and .

3-5. Find the limit.

3.

4. =

5.

6-9. Sketch curves with the given vector equations. Indicate with an arrow the direction in which  increases.

6.

7.

8.

9.

10-11. Find vector equations and parametric equations for the line segment  to .

10.

,

.

12. Show that the curve with parametric equation  lies on the cone .

We have .

13. Show that the curve with parametric equations ,   is

the curve of intersection of the surfaces  and .

We can see that the curve (red) is intersection of two surfaces (yellow, green).

14. Graph the curve with parametric equations

,

,

.

15. Show that the curve with parametric equations

passes through the points  and  but not through the point .

If , and if .

So, it passes through the points  and .

But, y-axis of the curve cannot be negative, so it does not pass through .

16-18. Find vector functions that represent the curves of intersection of the two surfaces.

16. The cylinder  and the surface .

Let . Then .

So,

is the vector function of intersection of two surfaces.

17. The planes  and .

Let . Then ,  .

Subtracting two equations, , so we have .

Substituting it to first equation, ,

so .

So,  is the vector function of intersection of two planes.

Drawing graph

18. The paraboloid  and the parabolic cylinder .

Put . Then ,  and .

So,

is the vector function of intersection of two surfaces.

We can see the answer is correct by drawing the graph of surfaces and vector function.

19. If two objects travel through space along two different curves,

it is important to know whether they will collide (Will a missile hit its moving target? will two aircraft collide?)

The curves might intersect, but we need to know whether the objects are in the same position at the same time.

Suppose the trajectories of two particles are given by the vector function.

,

for .

Do the particles collide?

Two particles collide means  has a solution.

But for second coordinate, ,

hence  has no real root, and hence there would be no collision.

20. Two particles travel along the space curves

.

Do the particles collide? Do their paths intersect?

First coordinate of two particles cannot be same, and hence there would be no collision.

To see which there is intersection, we draw graph of  and .

By rotating the resultant graph, we can see that there is one intersection.