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 tk, tR
(c) What surface does the equation r = 4(cos ui +
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 vectors a b c are coplanar if a b c = 0
1
2
1
1 0
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 linearly
independent? If p, q, s are position vectors of t
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 = 2i j + k
Answer
(a) (i)
c d = (5a b) (3a + 2b
= 15a a +
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 b + b c + c a|
|b c|
Answer
(a) if the plane passes thr
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 second line
(ii) The plane containing the second line an
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 is -1 then C1 , A1 , B1
are colinear.
B
Answer
@
C1
@
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 = 2i j + k
Answer
(a) c = 5a b and d = 3a + 2b
c d = (5a b
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 possible solutions if k = 0?
Answer
If kx + (x a)b = c
Th
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 lines, usually orthogonal, although not necessarily so.
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 point which moves so that its
distance from a fixed point
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-ordinates.
To answer this question we shall need to be able to
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 9th week, November 26th. No
rescheduling will be availabl
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 = adf6= 0 if and only id a,d, f are all 6= 0
df bf be cd
1
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 = det A det A1
therefore det A = det A1 = 1
1
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 cartesian graph. If we
have functions of a complex var
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 variables t; corresponding to each value of t
we have a
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 x2 = 2
In R we cannot solve x2 + 1 = 0
You have all done
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 parametric form of the equation BC and give its cartesia
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 determined.
Answer
(a) |r c|2 = a2 or (r c) (r a) = a2
r r 2
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 successive
reflections in each of three mutually perpe
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 10
!
2 3
BA =
14 19
!
16 21
2
A =
28 37
!
1 0
2
B =
3 4
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
=
=
=
=
11
0
0
2
5
12
19
t
Answer
For consistency t =
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 ?
Answer
If we perform this rotation successively 3 times we g
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 and verify that
the images are mutually orthogonal.
An