
Vectors
are one of the concepts in math whose name is so cool you imagine it
just HAS to be difficult. All the great concepts of math have names
that sound like Roman Gods or alien invaders; Vector in invader,
Calculon(ed: I think that's calculus) the destroyer, dot product the
accountant! These are the concepts whose legendary frightfulness keeps
children from running away from home, and whose reputation is largely
undeserved.
The opposite of a vector is a scalar, a number without a
direction, which represents only a magnitude
is called a scalar. 2 litres of milk. 100 kilometers (without saying
WHERE these kilometers are). 9.81 meters per second squared. degrees by
itself is a scalar.
A Vector is a number with a magnitude and a direction. If you've been
programming longer than a week, you've already dealt with them and you
don't even know it. A set of X, Y coordinates describing the motion of
an object can be considered a vector. A scalar whose negative sign
represents a direction can be a vector, like the distance from a
reference point. (ie. 1cm away from 0 and 1cm away from 0 meaning
different things). The concept of a vector is really simple.
More often than not though, when someone actually refers to a vector as
a vector, you're looking at a vector with a magnitude r and a direction
theta, which can be in radians or degrees. That's mostly because most
people don't think of vectors as any number with a direction and
magnitude. That's ok though, because these are fairly easy to do as
well:
The format for a 2d vector can be expressed in the following way:
r<theta
100cm<61 degrees
50km<11 rad
900 parsecs<11 gradians
A 3d vector has 2 angles; one for the X dimension, and one for the Z
dimension.
Vector math has been made complicated by enough people that I won't try
to. I'll just give you the basic formulas to work with vectors.
Basically, a vector can resolve to two formulas, keeping in mind that :
x = R * COS(theta in radians)
y = R * SIN(theta in radians)
That's all there is to it. If you were using vectors in a game to
control the movement of a car, you might add the above values to the
position of the car each frame, for instance:
car.x = r * cos(theta in radians) + car.x
car.y = r * sin(theta in radians) + car.y
And that's pretty much all there is to it. Adding another dimension
adds a Z to the mix, but luckily the axes are independant, so it just
means adding another variable to the mix.
Sometimes you'll want to add or subtract vectors. You should NOT do
this by adding the magnitudes and directions together! The best way to
do this is by changing them into X and Y coordinates, then adding the X
and Y coordinates together and turning the result back into the form
you want. Behold:
Here we convert both vectors into X, Y form:
X1 = r1 * cos(theta1 in radians)
X2 = r2 * cos(theta2 in radians)
Y1 = r1 * sin(theta1 in radians)
Y2 = r2 * sin(theta2 in radians)
Now we do the operation:
newX = x1 + x2
newY = y1 + y2
Finally, we change back to polar form:
newtheta = arctan(newy/newx) < that may need to be flipped...y/x or
x/y?
newR = (newX^2+newY^2)^.5
and we have our new vector in polar form, ready to go. It's all just
that easy! I hope this is enough to give you some ideas.
SJ Zero has
been using vectors for years without realizing it.
