harder GCSE vector questions

as far as I am aware these sorts of questions have only been posed by the Edexcel exam board in England (apologies if I've missed some from other boards)

they continue an erstwhile tradition - a general result seems to have been known to and used by Menelaus (~100CE, the logical one rather than the mythological one (maths rather than myths)) in his work on spherical geometry (but not involving vectors, obviously, because they were not invented until late in the 19th century)

a good focus on using vectors to prove (decent) results that can involve many variations of ratios

a powerpoint is here

180 degree rotations

if you download the powerpoint here the rotations are animated

rotate 180 about 1

rotate this (image) shape 180 about 2

rotate about another point 3

through 180 to end in the position shown above

students could produce their own examples

what do students notice if they connect together the four centres of rotation?

consider the effect of two 180 degree rotations about different centres

two rotations through 180 degrees seems to be a translation

what relationship is there between the translation vector and the two centres of enlargement?



a reason for this, approached vectorially:

grid moves

the purpose of these tasks is to involve students in simplifying expressions
although the letters are not used as variables, it is also a simple introduction to vectors
and an application of 'pythagoras' to find (direct, crow flies) journey lengths

can be used to ask about journeys, e.g. from R to D
from D to R etc.

consider why the journey stages sum to 0

could involve 'pythagoras' in working out the lengths

thanks to onlinedungeonmaster for this graphic

questions (9) and (10) contribute to an understanding of solving simultaneous equations

dividing line segments in a ratio

each of the shorter line segments is one third of the way along the line segment

also thirds
find the missing coordinates















this work links with the task 'jumping' where the triangle is at the mid-points of each side

jumping

this could be a construction question but is probably better (more accurately) done with a computer

start with any three points

start anywhere and jump ('leapfrog') over A - the same distance the other side of it (i.e. reflect in the point  'Geogebra' enables you to do this)

then, from where you end up, jump over B

and then C

keep on jumping, over A then B then C
what happens?

James Tanton explains what happens with this problem, using (kind of) vectors

how can you return to the start after just one 'cycle' - set of three jumps?

vectors

given that the parallelogram has sides with vectors a and b
heading away from the bottom left hand corner

what are the vectors for other lines in the tangram puzzle?