The maths of Vectors is not all that complicated, there is nothing new in terms of formulae to learn.
it all starts with
The Unit Vectors
we divide the 2 dimensional space up into a coordinate grid. You have probably used one like this before,
but it was probably labelled x & y ,
not i & j
it is practically the same, however we do not measure position, but rather the change in position
i & j vectors
so we can see that all 2 dimensional vectors need 2 values, one in the i direction and another in the j direction.
express the following vectors in terms relative to i & j unit vectors.
The distance left or right is the magnitude of i, its sign depends on direction, a positive i value is given to the right direction and a negative i value for vectors heading left
The distance up or down is the magnitude of j, its sign depends on direction, a positive i value is given to the up direction and a negative i value for vectors heading down
If we want to add them like we did in the graphical vectors section, we simply add the i vectors to the i vectors and likewise the j values to the j values
Be careful of the signs, like signs add and keep the sign
Opposite signs take away keeping the sign of the bigger number
The vectors in this grid are all labelled, write down a value in i & j for eachCalculate the following vector combinations from the vectors across.
a = -5i + 5j b = 3i - 4j c = 4i + 2j
d = 5i e = -2i - 5j f = 3i
g = -i + 4j f = 3i
To take a ij vector and translate it into magnitude and direction
how do we find the magnitude of the i & j component vectors
Magnitude of these vectors is simply found by using Pythagoras theorem
Many thanks to pythagoras and his theorm
but because hes not here to explain, let fowler explain...
So to get the size of the vector we use Pythagoas rule
Yes, Yes it should be i instead of x ..... but did you really notice ??
A vector is not complete without a direction (except for the Null vector)
Direction, having talked about direction, it can be easier to always have the North or South 1st followed by the angle to the East or West!
If we call teta that angle between the NS axis and the direction of the vector then we can find the direction every time
If you rearrange it then you will have to make a note of it ... work it out it is not hard.
- The first cardinal direction is the starting axis for your measurement,
- this is followed by the angle
- followed by the cardinal direction that the angle goes towards ....
really though, if you can use your trig identies (sine, cosine, tangent) you will have no problem converting from i & j to mag and dir of a vector, and from a mag and dir to i & j.
The 2 manners of representing the vector are interchangeable, and while it may be complicated at the minute, practice will make it second nature soon enough.
So now go and calculate the magnitude and direction of the following list of vectors
Composition of perpendicular vectors.
What we can do is multiply a vector by any value! The only thing that changes is the size of the vector, the magnitude (or if you are drawing it the length). The Direction DOES NOT CHANGE!
Well almost, if you multiply the vector by a negative value, then the direction get reversed. If you multiply by 2 then the vector gets twice as big. If you multiply by zero the vector becomes nothing!
parallel vectors is simple addition or subtraction of magnitudes depending on direction. Opposite directions take away!
A visual aid to describe the difference between distance and displacement
In exam questions they will ask you to give the velocity of the object after some calculations, you will only recieve full marks if you specify both the Magnitude and the Direction,