Cross product of vectors – Mathematics for Engineers – Vectors – TU Delft


Hi there! If you think about a way of multiplying vectors,
it turns out that you have several possible definitions: until now you have seen you can multiply
a vector and a scalar to obtain another vector. In the previous lesson you learned
about the dot product: an operation on two vectors
having a scalar as its outcome. Another name for the dot product
is the inner product by the way. This product is closely related
to orthogonal projections. In this video you will find out
about another operation on two vectors, whose result is another vector,
having some nice properties! This operation is called
the cross product or the outer product. Let’s think about the following question: Given two vectors u and v in three dimensional space, find another vector w
that is perpendicular to both of them. How can you find it? Not a difficult question to answer you will probably say: you know after all that this vector
then should have the property that the dot product with both vectors equals zero. For our example this means you have to solve
the following two equations: minus w1 plus 2 times w2 plus 2 times w3 equals 0 and w1 minus w2 plus w3 equals 0. Okay, this is not a problem, except that you
now have two equations with three unknowns! You may not know how to solve such a system yet, but in a course on linear algebra
you will learn how to do this. Furthermore you will discover that there are
infinitely many solutions. Because if you multiply any nonzero vector
that is perpendicular to the two given vectors, by a nonzero constant you get another vector
that is perpendicular to the two given vectors. And this you can do forever, you just change
the length of the perpendicular vector. One thing you could do to try to overcome
this non-uniqueness is to restrict your attention to unit vectors. But even then, you see that there are
still two possible directions in which the vector you are looking for can point. To choose one of the two directions,
there is nice rule called the right-hand rule. If the fingers of your right hand curl from u to v, then your thumb points in the direction of the
vector perpendicular to u and v that we will choose. This vector is the cross product of u and v. So if this is u and this is v, then the direction of the vector
you are looking for is this way. But if this is u and this is v,
I have to put my hand like this and you see that the vector points downwards! To get the point of this rule you should
really do the movement yourself! Fortunately, there is a procedure to compute
the cross product that is really straightforward. I call it the Amsterdam method, because
it uses crosses just as on the famous small pillars that can be found in Amsterdam. Let’s do an example with our vectors u and v: write down a table with two columns and six rows. In the first three entries of the first column
you put the vector u and in the next three entries you do this again. Then do the same thing for the second column
but now with the vector v. Then remove the first and last rows
and put three crosses in the table as shown. Now compute the three components
of the new vector w, using the crosses in the table. Each cross means upper left multiplied with
lower right minus lower left multiplied with upper right. If you practice this in the exercises
you will find out that this Amsterdam method gives a fast way to find the vector w. A quick check will reassure you that the vector w
is indeed perpendicular to both u and v. The cross product was introduced by the famous
British mathematician Rowan Hamilton in the early nineteenth century. In general the formula found from the
Amsterdam method to compute the cross product of the vectors u and v is shown here. I now challenge you to prove for yourself that the
vector w is indeed perpendicular to the vectors u and v! Given two vectors u and v in three dimensional space, you know how to find the direction
of the cross product using the right hand rule. The length of the cross product may seem
somewhat mysterious still. This length, though, has a clear and nice
geometric interpretation. If u and v are not scalar multiples of each other,
they actually lie in a plane: a plane is a two dimensional subset
of the three dimensional space. Then, you can define the parallellogram
spanned by u and v. This parallellogram then has an area; and guess what?! This area is equal to the length
of the cross product of u and v. Knowing how to compute
the cross product is one thing, the question of course is why on earth
would you want to be able to do such a thing? Well, there are numerous applications
in physics and in mathematics. In physics for instance you encounter it
in electromagnetism and mechanics on rotating bodies (like the earth). In mathematics the cross product is used
to compute the volume of a parallelepiped, and to determine the equations of planes
in three dimensional space. You will learn more about computing
these volumes and planes in the next videos. But first: do some exercises to compute cross products and study the properties of the cross product
listed in the text following this video!

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