Department of Mathematics and Philosophy of Engineering
MHZ3531 Engineering Mathematics IA
Assignment No.01
Due Date: 18/04/2018
Academic Year: 2017/2018
Instructions

Answer all questions.

Attach the cover page with your answer scripts.

Use both sides of paper when you are doing the assignment.

Please send the answer scripts of your assignment on or before the due date to the
following address.
Course Coordinator – MHZ3531,
Dept. of Mathematics & Philosophy of Engineering,
Faculty of Engineering Technology,
The Open University of Sri Lanka,
P.O. Box 21,
Nawala,
Nugegoda.
01. Define the vector product of 𝒂 and 𝒃, where 𝒂 and 𝒃 are vectors.
Explain the geometrical representation of the vector product.
Prove that the position vector of the point Z , which divides m : n the line joining the
nx + my
two points X and Y is
. Where the position vectors of the points X and Y
m+n
are x and y respectively.
(a). ABC is a triangle and O is a point inside of it. The line AO extended to L
which lies on the line BC such that BL : LC   : 1 .Similarly, BO meets the line
AC at M such that CM : MA   : 1 and CO meets the line AB at N such that
AN : NB   :1 .Consider OA  a, OB  b and OC  c .
1
(i). Find OL .
(ii). Show that a  c  b  a.
(iii). Deduce that   1 .
(b). OAB is a triangle such that OA  a and OB  b . The points P , Q and R lie on
the sides OA , AB and BO respectively such that
BR
OP
AQ
.
 ,
  and
RO
PA
QB


  1
 OAB
Prove that PQR  
   1  1  1 
.
Where PQR and OAB are area of the triangles PQR and ABC respectively.
02. Define the vector 𝜆𝒙 when 𝒙 is a vector and 𝜆 is a scalar.
(a).Prove that the equation of a straight line passing through a given point with the
position vector a and parallel to a given vector b is r  a  b , where  is a
parameter.
The points A and B have position vectors 2i  6 j  k and 3i  4 j  k respectively.
The line L passes through the point A and parallel to the point B . Prove that the
x  2 y  6 z 1


equation of L is
3
4
1 .
(b). Prove that the equation of a plane passing through a given point with the position
vector a and perpendicular to a vector n is r  n  a  n .
Find the equation of P be the plane perpendicular to the vector i  3 j  2k and
passing through the point with position vector 9i .
 2 1 1 


03. (a).Let A    1 2  1
 1 1 2 


(i). Find A2 and A3.
(ii).Using the characteristic equation of A prove that A 3  6A 2  9A  4I  0 ,
where I and 0 are the unit matrix and the zero matrix of order three.
().Hence find A-1.
().Show that A 6  6A 5  9A 4  2A 3  12A 2  23A  I  5A  I .
2
(b). Show that in a triangle ABC, if
1
1
1  sin A
1  sin B
2
sinA + sin A sinB + sin 2 B
1
1  sin C
0
2
sinC + sin C
then prove that ΔABC is an isosceles triangle.
 1 2 2 
1
04. (a).Verify that P   2 1
2  is an orthogonal matrix.
3
 2  2  1
(b).Consider the following system of equations.
x  2y  z  1
x  3y  2z  2  b
x  3y  ( 2  a  b) z  2
Where a and b are parameter.
(i). Write the augmented matrix of the above system.
(ii).For what values of a and b does the system have
().no solution
().exactly one solution
().infinitely many solutions. In this case please find all solutions.
(c).Find the set of solutions to the homogeneous system.
x1  x 2  x 3  x 4  0
3x 1  6x 2  x 4  0
 x2  x3  0
Hence, by first finding a particular solution, find the general solution of the
system.
x 1  x 2  x 3  x 4  2
3x 1  6x 2  x 4  5
 x 2  x 3  1
3