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.

This videos (Dot Products and Cross Products) helped me so much, I learnt more in 2 days than I did in the past 3 weeks when my semester started, Thank You so much!