Chapter 9 Vector Problem Set

1. Vector addition/scalar multiplication Kreyszig, 9th edition, problem set 9.1, problems 1-4, 7-10, 14-21

Find the component of the vector **v** with given initial point P and terminal point Q. Find |**v**|. Sketch |**v**|. Find the unit vector in the direction of **v**.

(a) P : (3,2,0)

(b) P : (1,1,1)

(c) P : (1,0,1.2)

(d) P : (2,-2,0)

2. Kreyszig, 9th edition, problem set 9.1, problems 32. (Think geometrically) If |**p**| = 1 and |**q**| = 2, what can be said about the magnitude and direction of the resultant. Can you think of an application where this matters?

3. Kreyszig, 9th edition, problem set 9.1, problems 36. (Think geometrically) (**Reflection**) If a ray of light is reflected once in each of two mutually perpendicular mirrors, what can you say about the reflected ray?

4. Kreyszig, 9th edition, problem set 9.1, problem 38, (a), (b), (d), (e), (g). (In each part, draw a figure of the geometric object, and denote each vertex of the figure by a distinct vector. Then use the basic laws of vector addition, subtraction, and scalar multiplication to find the required line-segments and points.) **Geometric Application** To increase your skill in dealing with vectors, use vectors to prove the followings.

(a) The diagonals of a parallelogram bisect each other.

(b) The line through the midpoints of adjacent sides of a parallelogram bisects one of the diagonals

in the ratio 1:3

(c) The three medians of a triangle (the segment from a vertex to the midpoint of the opposite side)

meet a single point, which divides the medians the medians in the ration 2:1.

(d) The quadrilateral whose vertices are the midpoints of the sides of an arbitrary quadrilateral is a

parallelogram.

(e) The sum of the vectors drawn from the center of a regular polygon to its vertices is the zero vector.

5. (Vector dot, cross, triple products) (Dot product) Kreyszig, 9th edition, problem set 9.2, problems 26, 27 Let a = [1, 1, 1], b = [2, 3, 1], c = [-1, 1, 0]. Find the angle between:

(a) **a**, **b**

(b) **b**, **c**

6. (Vector dot, cross, triple products) (Dot product) Kreyszig, 9th edition, problem set 9.2,31 note that the angle between two planes is defined as the angle between the vector perpendicular to each plane. (Planes Find the angle between the planes x + y + z = 1 and 2x - y + 2z = 0.)

7. Kreyszig, 9th edition, problem set 9.2, 34 (**Addition law**) Obtain by using **a** = [, ], **b** = [ , ] where

8. Kreyszig, 9th edition, problem set 9.2 42 parts (b) (c), (d). **Orthogonality** is particular important, mainly because of the use of orthogonal coordinates, such as **Cartesian coordinates**, whose "**natural basis**" (9), sec9.1, consists of three orthogonal unit vectors.

(a) For what values of are a = [, 2, 0] as b = [3, 4, -1] orthogonal?

(b) Show that the straight lines 4x + 2y = 1 and 5x - 10y = 7 are orthogonal.

(c) Find all unit vectors a = [, ] in the plane orthogonal to [4, 3].

9. (Cross/triple product) Kreyszig, 9th edition, problem set 9.3, problems 14, 6, 16, 17, With respect to right-handed Cartesian coordinates, let **a** = [1, 2, 0], **b** = [3, -4, 0], **c** = [3, 5, 2], **d** = [6, 2, -3]. Showing details, find

(a) (**c** + **d**) X **d**, c X **d**

(b) **b** X **c** + **c** X **b**

(c) (**b** X **c**) ... **d**, **b** ... (**c** X **d**)

(d) (**abd**), |(**abd**)|, (**bad**)

10. Kreyszig, 9th edition, problem set 9.3, problems 29. (Rotation) A wheel is rotating about the y-axis with angular speed . The rotation appears clockwise if one looks from the origin in the positive y-direction. Find the velocity and speed at the point (4,3,0).

11. (Areas and volumes) Kreyszig, 9th edition, problem set 9.3, problems 32 (**Parallelogram**) Find the area if the vertices are (3,9,8),(0,5,1),(-1,-3,-3),(2,1,4).

12. Kreyszig, 9th edition, problem set 9.3, problems 33 (**Triangle**) Find the area if the vertices are (4,6,5) ,(4,9,5), (8,6,7).

13. Kreyszig, 9th edition, problem set 9.3, problems 37. (First prove that the area of the parallelogram defined by any two vectors **a** and **b **equals **ab**. Further, the area of the triangle bounded by the two vectors and the diagonal of the parallelogram equals . In all cases, remember that the vectors need not start at the origin.) Find the volume of the parallelepiped determined by the vertices (1, 1, 1), (4, 7, 2), (3, 2, 1), (5, 4, 3)

14. Suppose and are two vector functions of the variable *t*. Prove the following formulae

(i)

(ii)

15. Scalar fields: Sketch the isoclines (curves where the scalar functions take the same values) for the following functions

(i) ,(ii) ,(iii)

16. Sketch the vector fields of the following vector functions

(i) ,(ii) ,(iii)

17. Give parametric representations for the following curves

(i) A circle with centre (3,0) and radius 2,

(ii) An arbitrary ellipse,

(iii) A helix passing through the points (0,0,1) and then (0,0,0),

(iv) The L-shaped figure formed by the two line-segments from (0,0,0) to (1,1,0), and from (1,1,0) to (1,1,1).

(v) Straight line passing through (4,0,1) and (0,2,4)

(vi) Intersection of x+y-z=2 and 2x-5y+z = 3

18. Find the parametric representation of the tangent unit-vectors of the curves in Problem 17.

19. Find the length of the curves in problem 17.

20. (i) Write the equation of the following curves in polar coordinates:

(a) A circle centred at origin.

(b) The Tropic of Cancer.

(c) The Greenwich meridian.

(d) Archimedean spiral.

(ii) (a) Write the equation of a helix in cylindrical coordinates.

(b) Describe in words the curve which has the following cylindrical coordinates .

21. Find the requested partial derivative by using the chain rule.

(i)

(ii)

(iii)

22. For the functions in 15, sketch the vector field corresponding to . What does this tell you about the relationship of the gradient of a function in two variables?

23. Find the maxima/minima of the following functions

(i)

(ii)

(iii) for the region

(iv) for the region

24. Find the directional derivative of f at P in the direction of a.

(i)

(ii)

25. Compute the divergence and the curl of the vector functions in Prob. 16.

26. Prove the formulae in Section 16 of the worksheet.

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