Calculus-Sec-11-2-Solution

11.2     Vectors

by SGLee - HSKim - JHLee

1-2. Determine .

1. (a)

(b)

(a) .

(b) .

2.

AB=<-2-(-1),1-2,-2-3>=<-1,-1,-5>

3-4. Find the sum of the given vectors.

3.

.

5-8. Compute  and .

5.

,

,

,

.

9. Determine a unit vector that has the same direction as .

The vector  has length ,

so the unit vector with the same direction is .

10. Find a vector that has the same direction as   but has length 6.

11. A clothesline is tied between two poles, 6m apart. The line is quite taut and has negligible sag.

When a wet shirt with a mass of 0.8kg is hung at the middle of the line, the midpoint is pulled down 6cm.

Find the tension in each half of the clothesline.

Let  and  represent the tension vectors in each side of the clothe line as shown in the figure.

Then  and  have equal vertical components and opposite horizontal components,

so  and . By similar triangles, .

The force due to gravity acting on the shirt has magnitude

Hence we have . The resultant  of the tensile forces counterbalances , so

and .

Thus, the tensions are .

12. The tension  at each end of the chain has magnitude 50N.

What is the weight of the chain?

13. (a) Draw the vectors , and .

(b) Show, by means of a sketch, that there are scalars  and  such that .

(c) Find the exact values of  and .

14. Let  and  in .

(i)  Plot the vectors  and .

(ii) Find scalars  and  such that .

15. If  and , describe the set of all points  such that .

Therefore the surface of a sphere with a center  and a radius .