Calculus III: The Cross Product (Level 1 of 9) | Geometric Definition


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.

7 thoughts on “Calculus III: The Cross Product (Level 1 of 9) | Geometric Definition

  1. I want to know.what is the cross product?? suppose if I put a force on something the result it "it'll move from one place to another" it's a result. then, I just want to know what is the practical use of this???

  2. How can two vector on same plane multiplying each other to give the third vector which is perpendicular to the previous vectors?

  3. Hello sir , I saw your many videos and I like your explanation.
    I was studying vectors , their products.
    But couldn't understand that why vector product is a vector quantity
    I hv many videos and read many books but everywhere has the same matter that this this..
    No one is telling why it is vector why dot is scalar
    Please make it clear if …

Leave a Reply

Your email address will not be published. Required fields are marked *