Dimensions and Vectors - a review
Dimensions
If we had a world of zero dimensions, there would only be one point. We would not be
able to go anywhere. To describe our position, we would not need any number since
there would only be one possible location.
If we had a world of one dimension (1-D), we could only go forward or backward, not
sideways. We would only need one number to describe our location with respect to some
starting point (origin). An example of this would be a line.
QUESTION: can you envision a 1-D universe that is finite with edges?
QUESTION: can you envision a 1-D universe that is infinite without edges?
QUESTION: can you envision a 1-D universe that is finite without edges?
(answers and examples will be shown in class)
If we had a world of two dimensions (2-D), we could still go forward or backward, but
we could also go sideways. We would need two numbers to describe our location with
respect to some starting point (origin). An example of this would be a surface.
QUESTION: can you envision a 2-D universe that is finite with edges?
QUESTION: can you envision a 2-D universe that is infinite without edges?
QUESTION: can you envision a 2-D universe that is finite without edges?
(answers and examples will be shown in class)
If we had a world of three dimensions (3-D), we could still go forward or backward and
side to side, but we could also go above and below. We would need three numbers to
describe our location with respect to some starting point (origin). An example of this
would be a volume.
QUESTION: can you envision a 3-D universe that is finite with edges?
QUESTION: can you envision a 3-D universe that is infinite without edges?
QUESTION: can you envision a 3-D universe that is finite without edges?
(answers and examples will be shown in class)
Can we extend this to four dimensions (4-D)?
Can you find an example of this?
Would this allow other “universes” to exist besides our own?
Vectors
Assuming we have more than 1-D, how do we work in this n-D universe? What do we
call a quantity that needs n numbers to describe its position? Are there some quantities
that need n numbers but others that need only 1 number? What about position, time,
mass, and velocity?
We name quantities that need only one number: scalars.
We name quantities that need n numbers: vectors.
There are many ways of assigning n numbers to a vector in an n-D space. The most
obvious way is to use rectangular: x,y,z. In 2-D, this reduces to x,y. Another common
way in 2-D is polar: r, .
QUESTION: how would you conveniently locate a point inside a doughnut?
QUESTION: is one system better than another, and is there one BEST system?
Adding displacement vectors:
Rectangular:
A = (4 bl, 3 bl)
B = (2 bl, -5 bl)
Result of adding A + B = (6 bl, -2 bl) in rectangular
Polar:
A = (4 bl, 3 bl) convert to polar:
r = [x2 + y2]1/2 = [(4 bl)2 + (3 bl)2]1/2 = 5 bl
 = arctan(y/x) = arctan (3 bl / 4 bl) = 36.9o
B = (2 bl, -5 bl) conver to polar:
r = = [(2 bl)2 + (-5 bl)2]1/2 = 5.39 bl
 = arctan(y/x) = arctan (-5 bl / 2 bl) = -68.2o
A+B = (6 bl, -2 bl):
r = = [(6 bl)2 + (-2 bl)2]1/2 = 6.32 bl
 = arctan(y/x) = arctan (-2 bl / 6 bl) = -18.4o
NOTE that
rA + rB = 5 bl + 5.39 bl = 10.39 bl does NOT equal rA+B = 6.32 bl and
A + B = 36.9o + -68.2o = -31.3o does NOT equal A+B = -18.4o .
Displacement vectors only add “nicely” in rectangualr form.
In PHYS 150, we had an experiment that showed that Force vectors could be added just
like displacement vectors, i.e., Force vectors added nicely in rectangular form, but not in
polar form.
Other operations with vectors:
Since subtraction is the inverse of addition, vectors can only subtract “nicely” in
rectangular form.
Since multiplication of a vector by a scalar is just multiple additions, vectors can
only be multiplied or divided by scalars “nicely” in rectangular form.
Since differentiation and integration are really division and addition of vectors,
these operations can only be done “nicely” in rectangular form.
Vector multiplications:
There are two basic ways two vectors can be multiplied together: the dot product
and the cross product. Both have nice examples in mechanics.
Dot product is necessary to understand the concept of work. If the force is in the
direction of the displacement, then work is done; if the force is perpendicular to the
displacement, then no work is done. Example with gravity: you do work when you lift
something up; you do not need any work to slide an object across a horizontal frictionless
surface. Therefore, only the component of the force that is parallel to the displacement is
involved, and this can be written as: W = F * s * cos(Fs), and we use the dot symbol to
show this: W = F  s where we use bold blue to show vectors. Note that work is a scalar
– it doesn’t have a direction and can be expressed by a single number.
Cross product is necessary to understand the concept of torque. To turn a screw
or bolt, you mush exert a force sideways at a radius from the center of rotation. The
component of the force parallel to the radius (moment arm) does not help turn the screw –
it may tend to shear the head off of the screw or bolt, but it doesn’t help turn it. We then
have for the torque:  = r * F * sin(rF), and we use the cross symbol to show this:  = r
x F . Note that torque is a vector – it does have a direction which by convention is
determined by the right hand rule: if you turn the screw or bolt in the direction of your
curled fingers on your right hand, the direction of the torque is in the direction of thumb.