Question
(a) Are the two lines x = 3y + 7, y = z 1 and x 1 = 3y 5 = 3z + 1
parallel? (Give a reason.)
(b) At what point does the line r = 12 i + uj + k uR cut the ellipse r =
sin ti + cos tj + 2 sin t
Question
(a) Obtain the value(s) of which make the vectors i j, 2i + j k,
i j + k coplanar.
(b) Show that the four points (5, 2 1), (6, 1, 4), (1, 3, 6), (3, 2, 1)
lie in a plane.
Answer
(a) The three
Question
The three vectors a, b, c are linearly independent.
(a) What does the equation
r = (1 + 2u v)a + (3v 6u)b + c
u, vR represent?
(b) When p = a + 2c, q = 2a b + 3c, s = 2b 3a 4c are p, q, s lin
Question
(a) For c = 5a b and d = 3a + 2b find c d when
(i) a and b are unit vectors at an angle
4
(ii) a and b are perpendicular with |b| = 2|a| = 2.
(b) Evaluate |c d| when c = i + 2j + 3k and d = 2
Question
(a) Show that the plane through the point a, b, c has the equation
rbc+rca+rab=abc
(b) Prove that the shortest distance from the point a to the line joining the
points b and c is given by
|a
Question
Two lines are given in space by the following equations.
x
y1
=
= z;
2
2
x+1=y2=
z+4
2
Find the equations of the following planes
(i) The plane containing the first line and parallel to the s
Question
Prove Menelanous theorem: If a line cuts the sides of a triangle ABC in
AC1 BA1 CB1
= 1.
the points C1 , A1 , B1 then
C1 B A1 C B1 A
Show, conversely, that if the above product of the ratios
Question
(a) For c = 5a b and d = 3a + 2b find c d when
(i) a and b are unit vectors at an angle
4
(ii) a and b are perpendicular with |b| = 2|a| = 2.
(b) Evaluate |c d| when c = i + 2j + 3k and d = 2
Question
When kx + (x a)b = c (b 6= 0) explain why x is of the form k1 c + b, k 6=
0 R. Hence solve kx + (x a)b = c
(i) k + a b 6= 0
(ii) k + a b = 0 and a c = 0 or a c 6= 0.
What can you say about po
Coordinate Geometry
In this section we shall discuss various co-ordinate systems in 2 and 3 dimensions and equations for a variety of curves.
Cartesian coordinates are formed by two sets of parallel l
Coordinate Geometry
Conic sections
These are plane curves which can be described as the intersection of a cone
with planes oriented in various directions.
It can be demonstrated that the locus of a po
Coordinate Geometry
Equations of Second degree
The problem dealt with here is the following, given an expression quadratic
in x and y, what curve does it represent in rectangular cartesian co-ordinate
Jonathan Jiang
A13098222
Section A
Ryan Cooper
Matlab Assignment 1
Exercise 1.0
a)
b)
c)
d)
Tuesday, Decemeber 1st
8:00a-8:50a
AP&M B349.
The alternate exam scheduling will close on Thursday of the 9t
Question
Show that the matrix
a b c
0 d e
0 0 f
is non singular if and only if a, d and f are all non zero.
Write down the inverse in that case. Check your answer by multiplication.
Answer
det A = a
Question
Prove that if a matrix A and its inverse both have all their elements integers,
then det A = 1
Answer
If A has all its entries integers then det A is an integer.
Ditto for det A1 1 = det A A1
Complex Numbers
Geometrical Transformations in the Complex Plane
For functions of a real variable such as f (x) = sin x, g(x) = x2 +2 etc you are
used to illustrating these geometrically, usually on a
Vector Algebra and Geometry
Differentiation of Vectors
Vector - valued functions of a real variable
We have met the equation of a straight line in the form
r = a + tb
r therefore varies with the real
Complex Numbers
History
The historical development of complex numberD.R.Green Mathematical
Gazette June 1976 pp99-107.
In N we cannot solve x + 2 = 1
In Z we cannot solve 2x = 1
In Q we cannot solve x
Question
The cartesian co-ordinates of the points A, B, C are (1, 1, 0), (1, 4, 6),
(3, 5, 7) respectively. Find
~ and AC.
~
(i) The components of AB
(ii) The direction cosines of line BC.
(iii) The p
Question
(a) Find the equation of a sphere centre c and radius a.
(b) Show that the equation of the tangent plane at a point d on the sphere
is
r d c (r + d) + k = 0
where k is some scalar to be deter
Question
The unit vector n
is along the bisector of the angle between the two unit
and d
0 . Show that
vectors d
n=d
+d
0 .
2(
n d)
Hence prove that a ray of light emerges parallel to itself after
Question
Let
A=
1 2 3
4 3 2
!
1 1
B=
2 3
0 1
Evaluate all possible products of pairs of the above matrices.
Answer
AA is undefined
!
5 10
AB =
10 15
!
14 9 7
AC =
16 1 8
3 1 1
BA =
10 5 0
4 3 2
B
Question
Let
A=
2 3
4 5
!
b=
1 0
3 2
!
.
Evaluate AB, BA, a, A2 B 2 , (A B)(A + B), (A + B)(A B).
Verify that
det AB = det A det B
and that
det(A B)(A + B) = det(AB ) det(A + B)
Answer
!
7 6
AB =
11 1
Question
Find the value of t for which the following system of equations is consistent,
and find the general solution in that case.
w + 2x y + 3z
2w + 4x + y + 5z
3w + 6x + 3y + 7z
7w + 14x + 2y + 18z
Question
A right-handed rectangular co-ordinate system is rotated through an angle
of 120 about the line x = y = z, Find the matrix A of the transformation
and show that det A = +1. What is A3 ?
Answe
Question
Let
1
1
1
3
2
6
1
1
1
.
A=
3
2
6
2
1
0
3
6
Verify that A is orthogonal. Suppose co-ordinates are related by x = AX.
Find the X equations of the images of the x1 , x2 and x3 axes