A displacement vector is simply an instruction that tells you how to move from one point to another. For instance, AB tells you how to move from point A to point B. This instruction might be something like:
- Go 2 miles due North
- Move 3 units parallel to the x-axis, and then 4 units parallel to the y-axis.
So all you need to know in order to describe the displacement vector AB is where B is in relation to A. So if A is the point (2, 10) and B is the point (4, 3), to get from A to B you would need to move:
- 2 units parallel to the x-axis (to change your x-coordinate from 2 to 4)
- -7 units parallel to the y-axis (to change your y-coordinate from 10 to 3)
(Can you see why it's -7 and not 7?)
Now there's a shorthand way of writing "Move 1 unit parallel to the x-axis" and it is to use the letter i (in bold type). This is called a unit vector. So if you want to say "Move 2 units parallel to the x-axis" just write: 2i.
In the same way, j means "Move 1 unit parallel to the y-axis". So to move -7 units parallel to the y-axis, you just write -7j.
If you want to move parallel to x, and then parallel to y, just add these shorthand things together with an ordinary plus sign. So:
- 2i - 7j means "Move 2 units parallel to the x-axis, and then -7 units parallel to the y-axis"
So that's how you would describe the vector AB if A is (2, 10) and B is (4, 3) (as we said before).
In the question you asked, A has a position vector 6i - j. That means you move to A from the origin (0, 0) by moving 6 units parallel to the x-axis, and then -1 unit parallel to the y-axis. If you do this you will simply get to the point (6, -1). B has position vector 2i + 2j, so it's coordinates are (?, ?).
(Did you say (2, 2)? You were right.)
So how do you move from A to B? Well, you've got to increase x by -4, to take it down from 6 to 2. And you've got to increase y by ??
(Did you say 3? You were right.)
So we need to move:
- -4 units parallel to the x-axis; and
- 3 units parallel to the y-axis.
In other words:
- ??i + ??j
(Did you say -4i + 3j? Right.)
So, that's AB: -4i + 3j.
I'll leave you to do AC and BC in the same way.
What about |AB|? This means the length of the movement from A to B. In other words, the distance AB. So draw a right-angled triangle, showing the movements of -4 units and then 3 units in the right directions, and join them to form the hypotenuse AB. Use Pythagoras to find the length AB. (It's the easiest right-angled triangle there is.)
(Did you say 5? Right.)
Do |AC| and |BC| in the same way.
What about the last part? Well, if triangle ABC is right-angled, then its longest side squared will equal the sum of the squares of the other two sides. So, you've just got to show that this is not so. (Easy!)
Sorry that's so long, but I hope it helps.