# 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.

please do more its useful

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???

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

Thanks it helps to me , love from INDIA

Very excellent explanation was more than wonderful

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 …

And why we choose smaller angleand R³ is 3D?