(3) Using components to describe vectors
We want a non-visual way of working with vectors
so we don't have to draw them to work with them.
Let's first think how we could do this if all the vectors
lie in the same plane so that we are in two dimensional space. If we
could describe each of these vectors in terms of some chosen reference vectors,
then we could work out how they affect each other by using algebra.
For example, in the drawing below I have chosen s and t as
my two reference vectors. From them we can build up a grid which makes it
possible to describe the position vectors of any points in the plane in
terms of s and t.
You can see that my three examples are described by p = s +
3t,
q = (5/2)s + t and r = - s + 2t.
As an example of working with these vectors, we have
q + r = {(5/2)s + t} + {- s + 2t}
= (3/2)s + 3t
which you can see will work geometrically on the grid.
We can describe any point in the plane using multiples of s
and t.
In other words,
if we choose any point P with position vector r then we can write
r = as + bt
where a and b are numbers
whose value depends on the particular position of P.
You can choose any pair of vectors to describe the positions of
all the points in this plane in this
way just so long as your two chosen vectors aren't parallel to each other.
Because neither s
nor t can be written in terms of the other, mathematicians call them
linearly independent.
(If you are visiting from finding vector
equations of planes, you can return here.)
In practice, there is a huge advantage in choosing s and t
so that they are perpendicular to each other, and making each of them
one unit in length. (Vectors with unit length are called
unit vectors).
We relabel them as i and j with i running along the
x-axis and j running up the y-axis.
Then we can write the position vector r of any point P as
r = ai + bj.
The particular numerical values given to a and b describe
where P is. I haven't shown the grid this time but it is just made up of
unit squares like ordinary graph paper.
The right-hand drawing shows this system extended into three dimensional
space by having a unit vector k running along a z-axis which is
perpendicular to both the x and y axes.
(In this drawing, I've chosen to keep x and y in the same positions
as the left-hand drawing.)
Now we have
r = ai + bj + ck
with particular numerical values of a, b and c for each r.
IMPORTANT We could have drawn the
z-axis in two ways to make it be perpendicular to the x and y axes.
In physical applications, we always choose the direction of the z-axis
so that x, y and z fit to the thumb, first finger and second finger of
the right hand. We then get what's called a
right-handed coordinate system.
You'll see that my axes fit this
rule and that your left hand
would give a mirror image.
Each of these axes can be extended in its negative direction also, so
that we can represent vectors such as
r = - 3i - 2j or r = 2i - j
- 5k.
Adding vectors is extremely easy if you know their components. All you have
to do is to find the total for each separate component.
For example, suppose you have 2 forces p and q with
p = 3i + j - 4k and q = 2i -
4j + 2k.
Then their
resultant r = p + q = 5i - 3j -
2k.
It is sometimes convenient to write the vectors i, j and
k in row vector form giving
i = (1, 0, 0) j = (0, 1, 0)
k = (0, 0, 1)
Any 3 dimensional vector can be written as a linear combination of these
3 vectors.
For example, (2, 3, -5) = 2i + 3j -5k =
2(1, 0, 0) + 3(0, 1, 0) - 5(0, 0, 1).
The next link extends this section with a practical example showing how
splitting forces
into components can make it easier
to write down equations to find them.
Next
If your trig is a bit shaky, you may like to
visit basic trig for working out vector
components first.
or back to the previous section
or return to the vector homepage.