Maths Warm-Up Exercises

The following exercises are to help you revise (or learn) the basic mathematics involved in this part of the course. Discuss any difficulties with the teaching assistant or teacher on this course, or other students on the course who have studied maths. The maths here is very simple, and in most other countries in the world it would be taken for granted that students on the 3rd year of any science course would have no difficulty with these topics. So if you have any major problems here, you are welcome to blame the British education system. However, this won't really help in the long run, since you can't get the British education system to write your programs or do your exams. So please --- take a trip to the library to see what you can find out.


1. What is a point? What is a vector? How are they related?

2. Find the dot products for the following pairs of vectors:

(a) (1,1,1) . (1,1,-1)
(b) (2,5,3) . (2,5,3)
(c) (1,0,0) . (10,1,5)

3. Find the length of the vector (2,5,3).

4. Let i = (1,0,0), j = (0,1,0) and k = (0,0,1). By considering the
meaning of cross product (*), write down the answers to:
i*i, j*j, k*k, i*j, j*i, i*k, k*i, j*k, k*j.

5. Using the result in 4. and the fact that (x,y,z) = x i + y j + z k
find an expression for p*q where p = (x1,y1,z1) and q = (x2,y2,z2).
Hence find the cross products for each of (a), (b), (c) in 2.

6. The equation of a plane is ax+by+cz = d. (Under what circumstances
will d=0?). Find a,b,c,d for each of the cases:

(a) The plane is the XY axis.
(b) The plane is the XZ axis.
(c) The plane is the YZ axis.
(d) The plane is parallel to the XY axis and of distance D from the origin.

Describe the geometric meaning of the planes:
(a) x=y
(b) x+y = 0
(c) x+y+z = 0
(d) x+y+z = 1.

7. Find the equation of the plane that passes through the following
three points:
 (1,0,0), (0,1,0), (0,0,1).

8. In general given the points (x1,y1,z1), (x2,y2,z2), (x3,y3,z3) find
the equation of the plane which passes through these points, by using
the plane equation
 ax+by+cz = d.

9. Let p0, p1, p2 be three distinct points in 3D. Then consider the
cross product n = (p2-p0)*(p1-p0). What does this vector mean
geometrically? Let p=(x,y,z) be any point on the plane formed by the
three points. What is the geometric relationship between the vector
(p-p0) and the vector n? Hence show that the equation of the plane
must be the dot product: n.(p-p0) = 0. By explicitly evaluating the
expressions, show that this is the same plane equation as you have
found in question 8.

10. Using the argument in 9., show that n must be normal to the plane,
and that in question 8 the normal to the plane is (a,b,c). What are
the normals to the planes given in question 6 (note that there are at
least two answers in each, case, give both)?

11. Evaluate the following product of a row vector and a matrix:

   (1, 2, 3, 4) | 1 2  3 4 |
		| 2 1  0 3 |
		| 2 0  1 3 |
		| 1 2 -1 1 |


12. Evaluate the following product of two matrices:

 | 1 2  3 4 | * | 2 1 3  4 |
 | 2 1  0 3 |   | 1 2 2  0 |
 | 2 0  1 3 |   | 1 1 1  1 |
 | 1 2 -1 1 |   | 2 3 1 -1 |

13. What is the transpose of the matrix in question 11?

14. What is the inverse of a matrix? How can you define it?

15. Suppose that A is a square matrix, and its inverse and transpose
are equal.  (Such a matrix is called orthogonal). Show that, for any
angle q, the following matrix is orthogonal:

  cos(t) sin(t)
 -sin(t) cos(t)

Suppose further that x and b are vectors of the appropriate dimension
and that xA=b.  How would you find x? In the example, suppose that b =
(1,2) find x.