The Cross Product Level 2
In the previous video we covered the geometric definition of the cross product. In this video
we will derive another method to compute the cross product between two vectors in space
by using their components. We can calculate the Cross product, A cross B directly if we
know the x, y and z components of vector A and vector B. We can calculate the components
of the cross product using a procedure similar to that for the dot product.
Let’s first determine the cross products of the unit vectors i-hat, j-hat and k-hat.
This will be pretty straight forward since the unit vectors i-hat, j-hat, and k-hat all
have a magnitude equal to 1 and they are perpendicular to each other.
Recall from the previous video that the cross product between two vectors in space can be
found by taking the product of the magnitudes of each vector times sine of the angle between
the vectors. Let’s use the geometric definition to find the cross product of all the possible
combinations between the unit vectors which include i-hat cross i-hat, j-hat cross j-hat,
k-hat cross k-hat, i-hat cross j-hat, j-hat cross k-hat, and k-hat cross i-hat.
The vector product of any vector with itself is equal to the zero vector, so i-hat cross
i-hat, j-hat cross j-hat and k-hat cross k-hat will all be equal to the zero vector. Next
let’s find the cross product between i-hat and j-hat, since the cross product between
2 vectors in space produces a vector that is orthogonal to these two vectors, it follows
that the cross product is equal to k-hat since this unit vector is orthogonal to i-hat and
j-hat and it is consistent with the right hand rule. In a similar fashion j-hat cross
k-hat is equal to i-hat and k-hat cross i-hat is equal to j-hat. The following diagram is
extremely useful for remembering the cross products between the unit vectors for example
i-hat crossed j-hat is equal to k-hat, when moving along the arrows clock wise the cross
product is positive, and when moving along the arrows counterclock wise the cross product
will be negative. Next we express vector A and vector B in terms
of their components and the corresponding unit vectors, then we go ahead and find vector
A crossed with vector B, we will find this by distributing the components. We then rewrite
the individual terms as follows, next we go ahead and compute the cross product of all
9 terms, 3 of the terms will be equal to the zero vector since these involve a unit vector
crossed with itself. The remaining 6 terms will be equal to a unit vector or the negative
of a unit vector depending on the order in which the unit vectors are crossed. The last
step is to group the terms that contain the unit vector i-hat, j-hat and k-hat together
doing that we obtain the components of the final vector which represents the cross product
between vector A and vector B. Notice that the cross product produces a vector with an
x, y and z component recall that the dot product produces a scalar and not a vector. This is
an alternative way of computing the cross product by using the components of two vectors
in space. Similar to the dot product we now have two distinct ways of computing the cross
product we can use the geometric definition or the component definition.
The component definition of the cross product is not an easy formula to remember. Luckily
there is a handy mnemonic that you can use to remember the cross product. In order to
make the component definition of the cross product easier to remember we use the notation
of determinants. If you are familiar with determinants from your previous math courses
you can go ahead and skip this section and jump straight into the determinant form of
the cross product. We will first review the determinant of a matrix.
Recall that the determinant of a square matrix (one that has equal number of rows and columns)
is denoted by enclosing the entries of a square matrix with vertical lines as follows. For
a 2 by 2 matrix the determinant is defined by the following expression, notice that we
multiply the diagonals of the square matrix and subtract the results, in the end we obtain
a numerical value and this numerical value is the determinant of the square matrix this
is also known as the determinant of order 2. For example the determinant of the following
matrices can be found by first multiplying the diagonal entries that move down from left
to right and subtract the product of the entries that move downward from right to left, notice
that the determinant can be positive, negative or zero.
Now to find the determinant of a 3 by 3 matrix or a determinant of order 3, we need to break
apart the 3 by 3 square matrix into three separate 2 by 2 square matrices. This can
be accomplished by using the method of cofactors or cofactor expansion. We first need to find
3 minors of the 3 by 3 matrix, recall that a “minor” is the determinant of the square
matrix formed by deleting one row and one column from some larger square matrix. Since
there are lots of rows and columns in the original matrix, you can make lots of minors
from it. For our purpose we will expand along the entries of the first row of a 3 by 3 square
matrix but you are free to choose which row or column you want to expand from. So given
the following 3 by 3 square matrix with the given entries we go ahead and find the 3 minors
by first generating three 2 by 2 square matrices by blocking the first row and systematically
blocking each of the 3 columns, by blocking row 1 and column 1 we obtain the following
2×2 square matrix, by blocking row 1 and column 2 we obtain the following 2 by 2 square matrix,
and finally by blocking row 1 and column 3 we obtain the final 2 by 2 square matrix.
Now that we have three 2 by 2 square matrices we find the minors by computing the determinant
of each 2 by 2 square matrix. We are not done yet. Remember the row that we used to expand
from? Well we need to make sure that we multiply each of the entries of this row with its respective
minor as follows. Lastly we need to determine the sign of the
cofactor by using the following expression, here i and j represent the row and column
number respectively for example, for the first cofactor we use the entry located at row
1 and column 1 so the overall sign is positive, for the second cofactor we use the entry located
at row 1 column 2 making the overall sign negative, and for the third cofactor we use
the entry located at row 1 and column 3 making the overall sign positive. Do not worry about
memorizing how one obtains the sign of the cofactor it will always be positive, negative
and positive as long as you are expanding from the first row which for our purpose it
will always be case. In the end a determinant of order 3 can be found by using the following
expression. For example the determinant of the following 3 by 3 square matrix can be
found as follows, we first find our 3 minors via cofactor expansion of the first row, doing
that we obtain the following 2 by 2 matrices, then it is just a matter of finding the determinant
of each 2 by 2 matrix and multiply this value with its respective entry from the first row
as follows, making sure we include the alternating sign positive, negative and positive. Simplifying
the expression we obtain the following for the determinant of the 3 by 3 matrix.
If we now rewrite the determinant of order 3 by replacing the first row with the standard
unit vectors i-hat, j-hat, and k-hat, and replace the second row with the components
of vector A and replace the third row with the components of vector B then vector A crossed
with vector B will be equal to the following determinant. Note the minus sign in the j-hat
component, the final expression for the cross product can be obtained by computing the determinant
for each 2 by 2 square matrix, with each 2 by 2 square matrix representing a component
of the cross product. Keep in mind that this 3 by 3 determinant
form is used simply as an aid to remember the formula for the cross product; it is technically
not a determinant because the entries of the corresponding matrix are not all real numbers
since we have the unit vectors in the first row. This is probably the easiest way of remembering
and computing the cross product. A second method of computing the cross product
does exist and it is slightly easier but many textbooks don’t go over this method because
it will only work for 3 by 3 determinants. This method modifies the determinant form
of the cross product by copying the first two columns onto the end as follows,
We now have 3 diagonals that move downward from left to right and 3 diagonals that move
downward from right to left. We essentially need to multiply along each of the 6 diagonals
and add the product of the diagonals that move downward from left to right and subtract
the product of the diagonals that move downward from right to left. In the end you simplify
by grouping the terms that contain the same unit vector. Notice that the j-hat component
of the cross product is written with the negative sign distributed.
Lastly let’s compare the component definition of the dot product with the component definition
of the cross product, many students get both of these vector operations confused. Notice
that the dot product is found by multiplying corresponding components and adding them together
which results in a scalar quantity, while the cross product is found by carrying out
operations with different components which results in a vector.
Alright in our next video we will go over a couple of examples and illustrate how to
find the cross product between two vectors in space.