Higher Mathematics Unit 3.1
Vectors
Contents
1.
1.
2.
3.
4.
5.
6.
Basic Ideas.
Some Important Results.
Collinearity in 3D.
“Section Formula”.
Scalar Product.
Some Points.
Basic Ideas
Points to cover: addition, subtraction, scalar multiplication, component form, position vectors,
magnitude of a vector, unit vectors ( i, j and k).
Examples on vector paths should be covered as should the distance formula.
Some teachers prefer to calculate AB , then AB , when the length of the line segment AB is
required.
Others prefer AB  ( x 2  x1 ) 2  ( y 2  y1 ) 2  ( z 2  z1 ) 2 .
With abler classes it is best to do both.
2.
Two Important Results
(a)
(b)
AB  b  a .
This is used a great deal. It is worth while going over the proof, giving more practice at
vector paths.
m  12 (a  b) . Go over the proof, which, again, uses vector paths.
In the 2D case, result (b) will be familiar to the pupils from their study of straight lines.
Working may be set out as follows:
A(3,4,7),
 2   3   1 
    

AB  b  a   0     4    4 .
  6   7    13 
    

1
3 2 40 7 6
5
M
,
,
 , M  ,2, .
2
2 
2
 2
2
3.
Collinearity
Pupils are often required to prove that 3 points are collinear.
B(2,0,6).
To prove that three points are collinear it must be shown that:
AB is a scalar multiple of BC (or AC ), AB is “parallel” to BC .
There is a point common to both vectors.(B here), and so the 3 points are collinear.
(i)
(ii)
A typical question is:
“ A(1,3,5), B(3,6,9), C (7,12,17). Show that A, B and C are collinear, and state the ratio in which B
divides AC.”
Working may be set out as follows:
 3 1  2
     
AB  b  a   6    3    3 
 9   5  4
     
 7   3  4
     
BC  c  b  12    6    6 .
17   9   8 
     
BC  2 AB . BC is a scalar multiple of AB ,  AB is “parallel” to BC .
B is a common point.  A, B, C collinear.
1
2
A
B
C
B divides AC in the ratio 1:2.
Note the usefulness of a sketch here; Pupils who fail to use sketches often get the wrong ratio, e.g.
2:1 here, instead of 1:2.
4.
“Section Formula”
A “common sense” approach is preferable, and working may be set out as follows:
(other approaches are possible)
“ A(3,4,9), B(17,11,39). P divides AB in the ratio 3:2. Find the coords of P.”
3
A
2
P
AP  53 AB
B
 17    3   20 

   

AB  b  a    11   4     15  .
 39   9   30 

   

 20   12 

  
AP  AB    15     9 .
 30   18 

  
3
5
3
5
 12 
 
A(3,4,9) and AP    9   P(9,5,27).
 18 
 
Questions requiring AB to be divided externally in a given ratio are stated in the following way:
P(1,1,1)
Q(6,9,13)
R
“Find the coords of R, given that PR  32 PQ. ”
 6  1  5 
     
PQ  q  p   9   1   8  ;
13  1 12 
     
 5   7 12 
   
PR  32 PQ  32  8    12 .
12   18 
   
 7 12 
 
P(1,1,1) and PR   12   R(8 1 2 ,13,19).
 18 
 
5.
Scalar Product
It is important that pupils are able to evaluate a.b by both a b cos and a1b1  a 2 b2  a3b3.
These results may be stated without proof.
Working may be set out as in the worked examples in the handout.
Some Important Points about Scalar Product
Y
(i)
b


a
cos 
X
a.b
ab
Z
cos 
XY. XZ
XY XZ
Calculations such as these come up every year. Therefore pupils must be highly competent at
them.
It is important to stress that the vectors must “point out” ( or “point in”) from the common vertex.
(ii)
Pupils must be competent at the following type of question:
a
b
c
“This triangle is equilateral, with side 2 units. Evaluate a.(a  b  c) .”
(iii)
Pupils must be able to use the Scalar Product to show that an angle is a right angle.
e.g.
Angle ABC is a right angle  BA.BC  0 .
C
B
A
This is best done by finding BA , BC and evaluating BA.BC . The conclusion should be properly
stated, e.g. “ BA.BC  0.  ABˆ C  90. ”
This result can be stated for perpendicular vectors.
Example
t 
 
For what value of t are the vectors u =   2  and v =
3 
 
Answer
 1
u.v = 2t – 20 +3t
 2
u.v = 0  t = 4
Question
The diagram shows a square-based pyramid of
height 8 units.
Square OABC has a side length 6 units.
The coordinates of A and D are (6, 0, 0) and
(3, 3, 8).
C lies on the y – axis.
a) Write down the coordinates of B.
b) Determine the components of DA and DB.
c) Calculate the size of angle ADB.
Answer
 1

2

3

4
2 
 
10 
t 
 
(1)
(2)
(4)
B = (6, 6, 0)
3 
 
DA =   3 
  8
 
3 
 
DB =  3 
  8
 
cosADB =
DA..DB
DA DB
6.

5


6
7
DA  82 , DB  82
DA.DB  64
ADB = 38.70
Some Points
(a)
Questions on vectors tend to be relatively undemanding. Therefore well-taught pupils
should be scoring highly on this topic.
Weaker pupils must be given a great deal of practice, for they, too, should score highly in
questions on vectors.
(b)
Sketches, either to obtain information from a complicated diagram, or to provide a
diagram in the first place, are essential. Encourage pupils to draw sketches.
(c)
Past papers contain many excellent questions; make use of them.
(d)
Area of triangle  12 bc sin A comes up often. Practice must be given.
Questions are of the type (with the coordinates of the vertices given):
Y
(i)
(ii)
X
Z
Find the size of angle X.
Calculate the area of the triangle.
For (i), use cos X 
XY . XZ
and for (ii), use area 
XY XZ
1
2
XY XZ sin X .
(e)
Collinearity proofs must be fully stated, as in article 3, above.
(f)
Pupils should realise that 3D coords and Position Vectors are just equivalent notations for
the same thing.
 3
 
e.g. P(3,4,5) and p   4  are equivalent.
 5
 
(g)
Practice must be given with i, j, k notation.
(h)
Pupils must be given practice at interpretation of sketches of 3D situations, where
coordinates of points shown in the sketch must be stated.
Weak pupils often need a great deal of practice at this kind of interpretation.
(i)
It is important that the pupils have practice at vector paths.
(j)
Vector methods are often very useful in proofs from “traditional” Geometry. The
following proofs can be done with high ability pupils:
“If quadrilateral PQRS is formed by joining the mid-points of the sides of quadrilateral
ABCD then quadrilateral PQRS is a parallelogram”
“The medians of triangle ABC are concurrent, meeting at the Centroid. The centroid
divides each median in the ratio 2:1.”
(k)
7.
Pupils need to understand the difference between component form and i, j, k form.
References/Resources
1.
2.
“H.H.M.” pp 205-223.
M.I.A. pp192 – 217