The Cross Product Level 1 In this video we will define a new vector
operation called the cross product also known as the vector product and sometimes referred
to as the Area product. In the previous videos we defined two ways in which you can multiply
vectors together, the first was scalar multiplication which produces a vector and the second was
the dot product which produces a scalar. The third way to multiply vectors is by multiplying
one vector by a second vector as to produce a third vector.
The first thing to keep in mind about these 3 separate ways of multiplying vectors is
that scalar multiplication and the dot product are defined for vectors in R squared and R
cubed. What makes the cross product different from the first two methods is that the cross
product is defined only for vectors in R cubed and not R squared. This is extremely important
to remember, I repeat the cross product is defined for vector in R cubed and not R squared.
When working with the cross product make sure you are using 3 dimensional vectors. Alright
with that out of the way let’s take a look at the geometric definition of the cross product.
The cross product is denoted with the classical multiplication symbol, and as the alternative
name for this operation suggests, the vector product is itself a vector. To define the
cross product between vector A and vector B we will start by drawing the two vectors
with their tails located at the same point. With both vectors aligned at a common point
vector A and vector B will lie in a common plane. Remember the cross product is defined
for vectors in R cubed so we are dealing with 3 dimensional vectors at all times. We define
the cross product to be a vector with a direction perpendicular or normal to this plane, which
in turn is also perpendicular to both vector A and vector B.
The magnitude of this third vector is equal to the product of the magnitudes of vector
A and vector B times the sine of the angle theta between vector A and vector B. We measure
the angle theta from vector A toward vector B and take the smaller of the two possible
angles. Similar to the dot product theta ranges from 0 to 180 degrees. With the cross product
defined this way the value of sine of theta will always be greater than or equal to zero,
this way the new vector will never have a negative magnitude, recall that the magnitude
of a vector is always a positive number or zero by using an angle between 0 and 180 degrees
along with sine of theta we are able to generate magnitudes that are positive or zero.
With the cross product defined this way vector A and vector B are going to be parallel when
theta equals 0 degrees and will be anti-parallel when theta equals 180 degrees, in addition
the magnitude of the new vector will be equal to zero. So it turns out that the cross product
of two nonzero parallel or antiparallel vectors is always equal to the zero vector. In particular,
the cross product of any vector with itself is also equal to the zero vector. We will
formally proof these properties in a much later video.
When vector A and vector B are parallel, the magnitude of the cross product will be zero
in this case it will be a minimum. When vector A and vector B are perpendicular, the magnitude
of the cross product will be a maximum. When theta is an acute or obtuse angle the magnitude
of the cross product between vector A and vector B will be a fractional portion or percentage
of the maximum magnitude. Now let’s talk about the direction of the
vector produced by the cross product. There are always two directions perpendicular to
a given plane, one on each side of the plane. How do we determine on which side will the
vector produced from the cross product point towards? By convention we use the “right
hand rule”, if we are trying to find vector A crossed with Vector B we can determine the
direction of the new vector by pointing your right hand fingers in the same direction as
vector A and then curl your fingers towards vector B. When curling your fingers make sure
you choose the smaller of the two possible angles since theta was defined to be an angle
between 0 and 180 degrees. Once you curl your fingers in the direction of rotation, your
straight thumb will then point in the direction of the vector produced when you cross vector
A and vector B with one another. A second way to think about the right hand
rule is by using 3 fingers, your thumb, index, and middle finger. The index and middle finger
will represent the vectors that are being crossed with one another, these are the vectors
that are located on the same plane. Your index finger will point in the direction of the
first vector in this case vector A and your middle finger will point in the direction
of the second vector in this case vector B, once you have aligned those fingers with the
corresponding vectors your thumb will point in the direction of vector A crossed with
Vector B. Now on the other hand if we are asked to find
the cross product between vector B and vector A we would obtain a totally different vector.
In this case we need to point our fingers in the direction of vector B and curl them
towards vector A, again making sure you choose the smaller of the two angles, you will discover
that your thumb now points in the opposite direction. The result is a vector that points
in the opposite direction to the cross product of vector A and vector B. This also tells
us that the cross product is not commutative! Vector A crossed with Vector B is not the
same as Vector B crossed with vector A. In fact they have equal lengths but opposite
directions. So keep this in mind when dealing with the cross product.
We can also illustrate the geometric interpretation of the magnitude of the cross product. If
the magnitude of vector A and vector B are represented by directed line segments with
the same initial point, then the vectors form adjacent sides of a parallelogram with base
equal to the magnitude of vector A, and altitude equal to the magnitude of vector B times sine
of theta, recall that the area of a parallelogram is given by the product of the base times
the altitude. Thus the length or magnitude of the cross product of Vector A with Vector
B is equal to the area of the parallelogram determined by vector A and vector B.
Lastly let’s compare both the dot product and cross product side by side. Many students
get these vector operations mixed up since both involve multiplication of vectors. The
first thing to remember is that both are operations involving multiplication of vectors the difference
is that the dot product will produce a scalar and the cross product will produce a vector.
In addition the dot product is defined for vectors in R squared and R cubed, and the
cross product is defined only for vectors in R cubed. The geometric definition of the
dot product uses the cosine of the angle between both vectors and the magnitude of the cross
product uses the sine of theta between both vectors and since the cross product also produces
a vector it will have a direction that is normal to the common plane between the two
crossed vectors. If n hat represents a unit vector normal to the plane containing the
crossed vectors then the cross product can be represented as follows.
Alright in our next video we will go over the component definition of the cross product.