All physical quantities we class as vectors are derived from DISPLACEMENT
so obey exactly the same rules for "adding" and "subtracting" as displacement
- that is geometry.
Notice that we multiply or divide by directionless quantities ( scalars
) which merely change the size of the original arrow but not the
direction - ie the scale of the arrow.
Different methods of adding vectors
There are many ways of doing the geometry of adding vectors. In the
end they give the same answers ( for most people ! ) so are entirely equivalent.
Simply adopt the way with which you are most comfortable.
SCALE DRAWING ( "Heads to Tails" )
This method has results as good as your drawings. The larger the scale
drawing the better. It is the method used in traditional navigation whether
by sailors, pilots or bushwalkers.
You need a fine pencil, ruler, eraser for your errors, protractor and
room for your drawing.
Technique
Decide your scale by looking at the sizes of your vector quantities. Typical
scales may be 1 cm = 1 unit or 10 cm = 1 unit.
Decide the order you wish to add the vectors and draw the first as accurately
as possible. Put an arrowhead on the end so you ( and everyone else ) knows
which way you are going. If you have directions relative to a compass,
draw North-South East-West lines.
Measure your angle to the next direction from the end of the first vector.
( If necessary you may have to redraw the N,S,E,W lines at the end of the
first vector.) Draw the second vector. Put an arrowhead on the end.
Now draw the total by going from the start to the end of the second vector.
We often put a double arrow in the centre of the total to make it clearly
different. Measure the length and the angles ( eg from the N,S,E,W directions
or the first vector).
Scale appropriately and quote the answer.
If you have many vectors, keep tacking them on to the end of each other.
The sum still goes from the start to finish.
ALGEBRAIC METHODS
When scale drawings are not accurate enough, it will
be necessary to use the algebraic rules for triangles.
METHOD 1
Using COSINE and SINE RULES for triangles
In the above vector triangle, angles and sides are labeled according
to usual ways - just spot the lower case letters to any old side but use
capitals for the angles opposite the sides.
COS RULE
No matter what the sides are
a2 = b2 + c2
- 2bc cos A or
b2 = a2 + c2 - 2 ac cos B
or c2 = a2 + b2
- 2abcos C
SIN RULE
a/ sin A = b/ sin
B = c/ sin C
Technique
Do a rough drawing of the vector sum - don't be too
precise
Label the sides of the resulting triangle with known
and unknown parts
Normally use cos rule first to find the missing side
length
Use sin rule to find the missing angle to give direction.
Check that the answer makes sense with respect to
your rough drawing.
Sometimes you may have to use sin rule first then
cos rule.
BEWARE:
sin ( 180 - angle ) = sin angle - can
muck you up
moving a vector to make the triangle plays havoc
with angles! The angle between vectors will change to ( 180 - angle
) when a vector is slid to addition position.
Example
If we add the two vectors above namely 25 ms-1
N300W to 50 ms-1 N300E then from the diagram
of the heads to tails, the angle opposite the sum = 1200 ( not
600! ) so the SIZE of the sum using cos rule
a2 = b2 + c2
- 2bc cos A
= 252 + 502
- 2x25x50xcos1200 = 625 + 2500 + 2500x0.5 = 4375
thus a = sum = 66.1ms-1
Angle using sin rule
a/ sin A = b/ sin
B , 66.1 / sin1200
= 50/ sin B thus sin B = 50 sin 1200
/ 66.1 = 0.6551
B = 40.90 The vector sum is 66.1ms-1
with an angle of N(40.9 - 30)0E
ie N10.9 0E
METHOD 2
USING COMPONENTS
This is used when many vectors are being manipulated.
It is not efficient for two vectors but great for multiple sytems and advanced
work.
