Vectors

Vectors explained. Reference to questions in c3&c4 book

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  • Created on: 24-06-10 01:33
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W8.1 Vector algebra
You may or may not have encountered vectors before.
They are quantities which have both magnitude and direction.
Think of them, if you like, as journey vectors,
representing a journey as a straight line, direct from its start to its finish.
We shall also use position vectors
[as the journey from the origin to the given point].
[In M1, we shall use vectors to represent force, acceleration, velocity,
momentum ...]
We shall develop our use of them, first, in two dimensions
and then stretch our results to three dimensions
[a stretch that is done very easily with vectors].
Notation
In geometry, AB can denote a line going from A to B
or the length of that line
[you have to tell from context which it means].
If you intend the line to be seen as a vector,
you must clearly indicate the direction in which the line is to traversed.
On a diagram, this is done by adding an arrow to the line.
You can also modify the name AB by putting an arrow over the letters:
with the name AB.
You can also refer to the vector with a single letter name
which must be in lower case bold: b.
[Upper case bold letters are reserves for the names of matrices ­
although these are not in our syllabus, we still ought to stick to the convention.]
In handwriting, you can't convincingly do bold,
so any single letter name for a vector must be underlined instead.
Another way of denoting a vector is with a column of numbers.
The vector shown here, which goes along 4 in the x direction and 3 in the y
direction, can be shown as vector .
If point A were the origin, then (4, 3) would be the coordinates of point B.
In you're A-Level work, a row like that will always be intended as a coordinate
pair and a column intended to be a vector.
CT Training 22/02/2010

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Beyond this course, you could well find vectors written as rows as well.]
Clearly, if A were the origin, then the vector could be used to describe the
position of point B.
In that case, it's known as a position vector.…read more

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For a vector , we define 2 as doubling each of the components, giving
.
On a diagram, this means preserving the direction of the line representing the
vector, but doubling its length.
One special example is what we mean by .
Just like with ordinary algebra, we understand this to mean 1 which has to
mean .
This is a vector which has the same size as the original,
but precisely the opposite direction.…read more

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Column vectors acquire a third component for the z direction: .
A new unit vector, k, along the z-axis turns i, j notation into i, j, k.
Everything else stays the same.
Exs
Perform each of these calculations in the i, j, k notation
and then repeat it using columns.…read more

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W8.2 Scalar product
Vector algebra so far has allowed us to add and subtract vectors,
to multiply a vector by a scalar
and to find the length of a vector.
What about some sort of multiplication?
There are, in fact, two types of multiplication with vectors,
one of which is a bit awkward and is not in your syllabus,
the other of which we do today.
Defn
a.…read more

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N.B.
When two lines cross, they form four angles:
two equal acute angles and two equal obtuse angles.
If the calculation gives a negative sign for cos ,
then that will give you the obtuse angle.
If you want the acute angle, you just subtract your answer from 180°.
Drill
Ex 9b
Test
AF3 ­ Polynomial division
W8.…read more

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Now for a + 2b:
Now a ­ b:
Notice that we've let the first two examples leave a trace.
Lastly, a ­ 2b:
The aim of this demonstration is to show that, whatever multiple of b you add
on to a, you will land up somewhere on the line through the end of a, whose
direction is determined by b.…read more

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These three are all for the same l, so we can put the three equal to each
other:
x­3= =
Without changing its meaning, we can reformat the x entry to be the
same layout as the others:
= =
This is the Cartesian equation of the line.
Notice that the denominators of the three fractions are the components
of the direction vector ­
that is, they contain all the directional information for the line.…read more

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We must first look at the two direction vectors and notice that they are
not the same and one is not a multiple of the other.
That means that the lines are not parallel, so we must proceed to
investigate whether they intersect.
If they do intersect, then we should be able to find values of l and m such
that the two equations give the same position vector.…read more

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Angle between two lines
The directional information in the equation of a line is entirely in its direction
vector,
so finding the angle between two lines is simply a matter of finding the angle
between their two direction vectors.
You should use the dot product like we did in the last lesson.…read more

Comments

daviesg

An in depth summary of notes on vectors, covering the basics up to scalar product.

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