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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…

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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.

Vector addition
If you imagine that vector a…

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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…

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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.

Perform each of these calculations in the i, j, k notation
and then repeat it using…

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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…

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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°.…

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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…

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These three are all for the same l, so we can put the three equal to each

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…

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l1: r = +

l2: r = +

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…

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and we could find its position vector
either by putting this value of in the equation for l1
or by putting this value of in the equation for l2.

Angle between two lines
The directional information in the equation of a line is entirely in its direction




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

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