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**furor celtica** 1. If **p** = 2**i** + 3**j**, **q** = 4**i** – 5**j** and **r** = **i** – 4**j**, find a set of numbers f, g and h such that f**p** + g**q** + h**r** = 0. Illustrate your answer geometrically. __Give a reason why there is more than one possible answer to this question.__

2. If **p** = 3**i** - **j**, **q** = 4**i** + 5**j** and **r** = -6**i** + 2**j**

a. can you find the find numbers s and t such that **q** = s**p** + t**r**

b. can you find numbers u and v such that **r** = u**p** + v**q**?

__Give geometrical reasons for your answers. __

As you can see these are pretty basic vector problems. I didn’t have any difficulties with the strictly algebraic part of these questions, and my answers are as follows:

1. f = 1, g = -1, h = 2 ( I am aware that any multiples of these numbers would do as well.)

2. a. No.

b. Yes: u=2, v=0

However I’m not sure what they mean by, ‘giving reasons’. This sounds very vague to me, and I’d like to know what sort of answer I’m supposed to be giving, along what lines this ‘proof’ is supposed to be.

Here is my attempted explanation for 2.a.:

**r** is a multiple of **p**, which means that the result of s**p** + t**r** must be a multiple of **p**; **q** is not a multiple of **p** therefore numbers s and t do not exist for which **q** = s**p** + t**r**. (At this point I remembered I was supposed to give ‘geometrical reasons’ (what is that supposed to mean anyway?) so I added this)

Geometrically, points (0i, 0j), (3i, -j) and (-6i, 2j) are collinear: s**p**, t**r** and **q** will not meet thrice and a triangle will not be formed.

As you’ve noticed I’m not really sure where I’m going with this explanation (especially towards the geometrical part, that was just embarrassing to post), so if I had a few tips on what sort of ‘reasons’ I’m looking for in all three questions that would help a lot. Cheers