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Practice Questions
Clement Radcliffe, Contributor
Last
week's lesson presented relevant information on Vectors in general. The Position
Vector and some problens involving vectors were also reviewed. While there
are other methods we concentrated on the use of vector components in the Cartesian
diagram to evaluate vector problems. We
will now look at the Homework.
In
the diagram above, A and B are points such that  =
a and  = b.
The point P (not shown) in such that = 1/2 a + b. (i)
Write  in the form
 (ii)
Determine the length of OP. SOLUTION (i)
From the diagram, the coordinates of A = (6,8) and B + (5,11). It was illustrated
in last week's lesson that if the coordinates of A are (6,8), then the position
vector =  ie.
position vectors 
You
therefore add both answers. (ii)
Since
+
= c 
As
|c| is the length of vector c, 
We
will now direct our attention to MATRICES. MATRICES
REMINDERS
- A
matrix is a rectangular array of numbers, for example,
 - The
order is a 2 x 3 matrix, with 2 x 3 representing the order.
- The
order identifies the number of rows and columns respectively
Other
examples of matrices follow.
Please determine the respective orders of the
following: 
I
do hope that your answers are:-
(a) 2 x 2
(b)
2 x 1
(c) 1 x 3 An
analysis of the types of problems set by CXC would suggest that the following
are the usual types set with respect to matrices. - Application
of the Arithmetric Operations to matrices.
- Use
of matrices to solve simultaneous equations
- Matrix
Transformation
In
all these areas, the methods involved are relatively straightforward. The students
who take time out to understand, study and practise them, experience very little
difficulty. The areas providing most difficulty are: - Multiplication
of matrices
- Determining
the Inverse of a matrix
- Matrix
Transformation
I
do hop that you realize that the coordinates of the point P are (8, 15). (ii)
Using the formula for length" OP²
= x² +
y² {Using
Pythagoras' theorem}
= 8²
+ 15²
= 64 + 225 = 289
ie. length of OP =  289
= 17 I
do hope that you had no difficulty in understanding the above. If this is the
case, let us attempt another example. 
Using
the graph (i)
Express each of the position vectors
and
in the form . (ii)
Determine the vectors:
(a) 3,
(b) 2 hence
determine (c)
3
- (-2) (iii)
If OA + OB = c, show that |c| = 34 SOLUTION (i)
The coordinates of A and B respectively are (4, 3) and (-2, 3). ie.
the position vectors are
=  and
 Note
that the coordinates of A and B were used to determine the position vectors
and
You could also have read off the components directly from the graph. (ii)
(a) Given that =
then 3
=  Answer
is (b)
Given that
=
then - 2
=  Answer
is (c)
3
- 2
=  BE
WARNED, BE PREPARED Please
spend adequate time to ensure that you are comfortable with them. We will now
review Application of Arithmetic Operations to matrices. MATRIX
ADDITION Only
matrices of the same order may be added or subtracted. Corresponding elements
are added or subtracted. 
Find
(a) A + B
(b) B - A
(c) A + C SOLUTION 
(b)
On your own, prove that B - A is  (c)
A + C is not possible as the orders are different. Please
attempt the following for Homework. 1.
Given the vectors: 
Evaluate;
(a) A + B
(b)
A - 2B 2.
Given that: 
Find
the values of x and y.
As
the day of the exam approaches, I am still demanding that you find appropriate
exercises from your text books and past papers and practise, practise, practise.
It is bound to pay off in the long run. Clement
Radcliffe is the principal of Glenmuir High School in May Pen. | 
