What are vectors?
By becky on Sunday, June 4, 2000 -
05:16 pm :
Hi, I need to know about scalar quantity and vectors for my
GCSE maths! Can anyone help!
By Neil Morrison (P1462) on Monday,
June 5, 2000 - 05:25 pm :
Vectors are basically directions. You've probably seen them in
letter-form (4i + 3j ) etc if you're in England.
but you may be more familiar with the column form:
The basic idea is that vectors show direction along
different (perpendicular) dimensions. If you use lettering, you
may know that i represents x and j represents y.
They are both unit vectors, ie:
|
i= |
æ
ç
è
|
|
ö
÷
ø
|
, j= |
æ
ç
è
|
|
ö
÷
ø
|
|
|
That is why we express other vectors in terms of them. So
a vector
means 5 to the right and -2 upwards, which means 2 downwards.
(This is in 2D. In 3D, these correspond to directions in a
horizontal plane, with height measured along the z dimension; the
corresponding unit vector is k . But you're probably only be
working in 2D just now.
Anyway, position vectors don't give direction as such, they are
like coordinates. If
then you can say the x,y-coordinate is (3,10) if the same
origin is used. So they are sort of different.
Suppose we had a line from A to B (the from part is quite
important), and A and B have position vectors as below:
|
a= |
æ
ç
è
|
|
ö
÷
ø
|
, b= |
æ
ç
è
|
|
ö
÷
ø
|
|
|
Now
we say the direction vector of the line is the direction you go
to get from A to B. To find this, you imagine you are not going
straight to B, but you are going along the x direction as far as
B, and then along the y direction towards it. So you're taking a
right-angled detour. The point of this is to see how far you have
to go in each direction, because vectors treat each direction
separately. So in the x direction you have to go from 1 to 6,
which is 5, and in the y direction you have to go from 4 to 3,
which is -1.
So the vector
In other words, AB = 5i - j . Do you understand
so far?
In general: vector A to B = b - a , where b
and a are the position vectors of each.
Now length. Remember we took a detour instead of going straight?
Well the length (or magnitude) of the vector is the distance we
would have travelled if we had moved straight. You can work this
out using Pythagoras, because you worked out the two legs of the
right-angled-triangle, and you want to find the hypotenuse.
So the two legs are 5 and -1 (ignore the -, it just shows we're
going backwards, and the formula works the same) so Pythagoras
gives sqrt(26) as the length of this vector. We call this
|AB|
Generally:
If AB = pi + qj , then
|AB| = sqrt(p2 + q2 ) just like pythagoras.
I hope all this is not confusing.
I won't do scalar product (not yet at least).
Neil M