Vectors in Three Dimensional Space
In single variable calculus,
or Calc 1 and 2, we have dealt with functions in two dimensions, or R2.
In multivariable calculus, we will need to get accustomed to working in three dimensional
space, or R3. Most of our notation and calculation will be the same,
but with the extension of an added variable, z.
The extended Cartesian graph now looks as follows
The xy plane is now layed out horizontally extending towards you and into the screen.
The equation of this plane can be described by the equation z = 0, where the function
does not have a height.
The xz plan is vertical and also extended out and towards you. Functions that exist
only on this plane have a y value of 0.
The yz plane has now taken the place of the xy axis in 2D and is extending up and
down as well as left and right. This plane has an x value of 0.
Figures in Three Dimensions
We can still plot 2 dimensional shapes in 3 dimensions, all we need is to set any
of the axes to 0.
The formula for a circle in the xy plane would be
Notice that when z is 0, it is the same equation for a 2 dimensional circle. When
z does not equal 0, the graph will form a sphere.
The graph of a circle in the xy plane forms a cylinder when z can take on any value
Let’s graph x = 3 in one, two, and three dimensions. Here is x = 3 in R1
Since x is the only dimension, the image is just a point on a number line.
x = 3 in R2
Because y can take on any value when x is 3, the image is a straight line.
x = 3 in R3
Here, y and z can take on any value, so the image is the yz plane at x =
3.
Notice that first we have a point, then a line, and then finally a plane.
Vectors in Three Dimensional Space
Let’s graph the vector v = <3,5,7>.
We use the same method of plotting points and lines in three dimensions as we did
on the two dimensional xy plane
The Distance Formula
The Distance Formula for finding the distance between two points is just extended
from our formula in two dimensions.
The magnitude of vector v would be
Vector notations, equalities and operations from
Introduction to Vectors apply to 3D space, just with an added z axis. There
is, however, one operation that requires the vectors to be in 3 dimensions.
The Cross Product (Vector Product)
The Cross Product as another way of multiplying vectors. Unlike the Dot Product,
the Cross Product finds the vector that is orthogonal (perpendicular in 3D)
to both vectors, so we can only take the Cross Product in three dimensions. The
result is also going to have size and direction, which makes it a vector.
If we have two vectors u and v, the cross product is defined as
This is the same as the dot product, but we take the sin of the angle and multiply
it by n, which is a unit vector that provides us with the direction because it is
orthogonal to both u and v. We will soon see how to ultimately find the direction
of the result of the cross product using the right hand rule.
Let’s look at arbitrary vectors u and v.
If we decontruct the formula for the cross product, we can understand what it actually
does. Taking the last part of the cross product, we get
This basically is taking the direction vector of v that is perpendicular
to u multiplying it by the magnitude of both v and u. Unlike the dot product
where we project v onto u, we project v onto the perpendicular of u. Then we multiply
this scalar by n, which is the unit vector that is perpendicular to both
v and u (this is why the vector product only works in more than two dimensions).
If we look at the vectors, there are two directions that are perpendicular to both
of them – into the screen and out of the screen. This is where the “Right Hand Rule”
comes into play.
The Right Hand Rule
First, point your index finger in the direction of u. Then, point your middle finger
in the direction of v. Now, wherever your thumb is pointing is where the direction
of resulting vector of the cross product is pointing.
In this case, your index finger should be pointing up, your middle finger should
be pointing to the right, and your thumb should be pointing towards the screen.
This means that the resulting vector is pointing in the screen, orthoganal to both
vectors u and v.
The cross inside the circle means that the vector is going inside the screen. We
denote a vector going out of the screen by a circle with a dot in the middle. This
is because the tip of an arrow makes a dot as it is coming at you and leaves an
“X” from behind.
Here is another view so we can see the vector clearly. To make this view, we rotated
the graph ninety degrees forward where u is in the foreground and v is in the background.
We can see that n is the unit vector giving the cross product direction.
Before we multiplied by n, the result was a scalar (just like the result of a dot
product). When we multiplied by n, we multiplied by the unit vector orthogonal to
both original vectors.
The Right Hand Rule is a convention that was used to standardize the cross product.
The cross product is used in many fields within physics, such as finding torque
on an object and calculating magnetic fields of particles. Since we don’t normally
do this on a day to day basis, the cross product is a bit counterintuitive in its
motivations. We must remember that the order of vectors in the cross product matters,
and we must make sure we remember the right hand rule.
Note that if we take the cross product of parallel vectors, the result with be 0
because there is no angle between them (sin of 0 is 0).
The cross product can be represented in the form of unit vectors, similar to the
dot product.
This form is a bit harder to remember, but can be useful if we don’t know the angle
between vectors.