The Cross Product Level 7
In this video we will go over our first application of the cross product.
In the previous video we used the cross product to find the area of a parallelogram formed
by two adjacent vectors.
We can extend this idea to three dimensional space and use the cross product to determine
the volume of a parallelepiped (a polyhedron whose faces are all parallelograms).
If vector A, vector B and vector C are in R cubed then vector A dotted with the cross
product of vector B and vector C is called the Scalar Triple Product or Triple Scalar
Product.
The scalar triple product can be written in determinant form by using the components of
the vectors as follows.
Note that the value of the determinant is multiplied by negative 1 if two rows are interchanged.
After two such interchanges, the value of the determinant will be unchanged.
So, the following triple scalar products are also equivalent.
If vectors A, B and C do not lie in the same plane, the triple scalar product can be used
to determine the volume of the parallelepiped with vector A, B and C as adjacent edges.
In this figure the area of the base is a parallelogram and can be found by computing the magnitude
of the cross product between vector B and vector C.
If we were to draw the vector obtained by crossing vector B and vector C we obtain a
vector that is orthogonal to the plane containing these two vectors.
Now if we use theta to represent the angle between vector A and the cross product between
vector B and vector C, then the height h of the parallelepiped will be equal to the magnitude
of vector A times the absolute value of cosine of theta.
Note that we must use the absolute value of cosine of theta instead of cosine of theta
in the case when theta is greater than pi over 2, since we want the height to be a positive
number.
The volume of the parallelepiped is going to be equal to the area of the base times
the height, substituting these expressions for the area of the base and the height, we
obtain the following expression for the volume of the parallelepiped, recall that this is
nothing more than the geometric definition of the dot product.
Making sure we take the absolute value since volumes are positive values.
We can also use the scalar projection of vector A onto the vector produced by crossing vector
B and vector C. Again making sure we use the absolute value of the scalar projection since
the height will be a positive number.
The magnitude of the cross product cancels out and we are left with the same expression
for the volume of the parallelepiped.
A natural consequence of the scalar triple product is that the volume of the parallelepiped
is 0 if and only if all 3 vectors are coplanar.
That is, if vectors A, B and C have the same initial point then they will lie in the same
plane if and only if the scalar triple product is equal to 0.
The last thing to keep in mind is that the expression obtained by crossing vector A with
the cross product of vector B and vector C is called the Vector Triple Product or Triple
Vector Product.
This expression is very useful for simplifying vector calculations in physics.
In this course we will mainly use the scalar triple product and not the vector triple product.
Alright let’s go over a couple of examples that makes use of the scalar triple product.
Find the volume of the parallelepiped whose edges are vectors A, B and C.
In this problem we are given three vectors in component form.
These vectors represent the edges of a parallelepiped.
We can easily find the volume of the parallelepiped by computing the scalar triple product.
So let’s write the scalar triple product in determinant form as follows.
Next let’s go ahead and find the minors by expanding row 1.
We then find the determinant of each 2 by 2 matrix and simplify, doing that we obtain
the final answer equal to 82 cubic units.
Alright let’s go over the next example.
Find the volume of the parallelepiped with vertices P, Q, R and S and vectors PQ, PR
and PS as adjacent edges.
Alright similar to the previous example we are asked to find the volume of the parallelepiped.
We are not given the components of the vectors so we first need to find them.
Subtracting the appropriate coordinates from one another we obtain the following components
for vector PQ, PR, and PS.
Next we go ahead and compute the scalar triple product.
So we set up our determinant and find the minors as follows.
Then we find the determinant of each 2 by 2 matrix and simplify the expression, doing
that we obtain negative 3.
Recall that the volume has to be a positive number so we take the absolute value obtaining
3 cubic units as the final answer.
Let’s try the next example.
Use the scalar triple product to verify that vector A, B and C are coplanar.
Here we are given 3 vectors in standard unit vector form; let’s rewrite them in component
form.
We are asked to determine if the vectors are coplanar we can easily determine this by calculating
the scalar triple product and if the result is equal to zero then the vectors are coplanar.
So let’s set up our determinant as follows.
Next let’s find the minors of the matrix, then we compute the determinant of each 2
by 2 matrix, simplifying the expression we obtain 0 as our final answer.
This means that the volume of the parallelepiped is equal 0 this result confirms that all three
vectors are coplanar.
Alright let’s go over the final example.
Use the scalar triple product to determine whether the points P, Q, R, and S lie in the
same plane.
Alright this is very similar to the previous example.
We are given the coordinates of 4 points instead of providing us with vectors.
In order to determine if all 4 points are coplanar we need to generate 3 vectors for
which all have a common initial point but have distinct terminal points.
For this problem I will use point P as the common point this way I can generate vector
PQ, PR, and PS.
We first need to find the component form of each vector this can be done by subtracting
the terminal point from the initial point for each vector, doing that we obtain the
following components.
Next we need to calculate the scalar triple product so we set up our determinant and find
the minors, doing that we obtain the following, computing the 2 by 2 determinants and simplifying
we obtain the final answer equal to 0.
All 3 vectors are coplanar since the volume of the parallelepiped is equal to zero.
Since these 3 vectors also include all four points we can also conclude that points P,
Q, R and S also lie in the same plane so they are coplanar.
Alright in our next video we will go over our final application and learn about torque.