history of mathematics - National STEM Centre

Performing this step three successive times gives a
step of one and a half times the standard length and with
direction to the left. Reversing gives a step of one and a
half times the standard length and with direction to the
right. It can be expressed in symbols as
Activity 11.10, page 145
1 a You take a step of one unit to the right (that is
labelled by i) and multiply it by the scalar 2. This gives
you a step of length two units to the right. The
incorporation of the minus sign (equivalent to
multiplying by the scalar -1 ) reverses the direction. Thus
-2 x i represents a step of 2 units to the left.
b i + j represents a step of length A/2 inclined at an
angle of 45° to each of the basic steps i and j. You will
find the details in the hints to this activity.
c You should consider -(^ x i) and -(•} x j) separately
first. (^ x i) represents a step to the right with size one
third of the size of the basic step. Then incorporating the
minus sign changes the direction so -(-j x i) represents a
step of one-third to the left.
(| x j) represents a step upwards with size one and a half
times the size of the basic step. Then incorporating the
minus sign changes the direction so -(4 x j) represents a
step of one and a half downwards.
The combination of the two produces a step as shown in
the figure below.
step ,
downwards
step to left
outcome
Activity 11.11, page 145
1 a / x / x i represents the application of the turning
operation /, not once but twice, to the step i. Applying /
once to i gives j as outcome. If you apply / again then
you must apply it to the outcome j to obtain -i. So
7x/xi = -i.
b In a similar way, /x/xj= /x-i = -j.
c You can express the results of parts a and b as
14 Answers
7 2 x i = -i and 7 2 x j = -j. Therefore 72 has the effect
of incorporating a minus sign into the basic steps and so
it behaves in the same way as the scalar —1 .
2 In each case you multiply out the brackets, and then
you treat each subsequent term separately.
In Activity 11.10 you showed that i + j is represented by
a step of length V^ inclined at an angle of 45° to i.
The outcome here is the composite step j - i . You can
show that this is a composite step of length A/2~ inclined
at an angle of 45° to each of -i and j. In each case you
can see that the effect of applying the operator (1 + /) to
the basic steps is to turn the step through an angle of 45°
in an anti-clockwise direction. (1 + /) is a turning
operator.
(2-/)xi = 2xi-(/xi) = 2xi-j
In these two examples the operator (2 - /) is acting on
the basic steps i and j in turn. You can see the effect of
the operator (2 - /) on each of the basic steps by drawing
diagrams showing both the outcome and the step with
which you started. These are shown in the figure below.
From these you can deduce that (2 - /) is a composite
operator consisting of both a turning, or rotation, and an
enlargement. In each case the step with which you started
has its size enlarged by a multiple of V5 and at the same
time it is turned through an angle 6 in a clockwise
direction where tan 6 = 0.5 . The scalar term in the
composite operator represents the scaling factor (in this
case an enlargement) and the scaling, either a stretch or a
contraction, will take place parallel to the step to which
the operator is applied.
3 You can think of 2 x /as being the same as / + /, so
(2 x 7) x i = (7 + 7) x i = j + j = 2 x j. Thus the effect of
multiplying the operator by a number is to multiply the
original outcome by that same number. In this case since
the number is greater than 1 a stretch as been applied
upwards. So the scalar factor multiplying the turning
183