Because the cross product of two vectors is a vector, it is possible to combine the dot product and the cross product. The dot product of a vector with the cross product of two other vectors is called the triple scalar product because the result is a scalar.
Definition
The
triple scalar product of vectors
$\text{u},$$\text{v},$ and
$\text{w}$ is
$\text{u}\xb7\left(\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{w}\right).$
Calculating a triple scalar product
The triple scalar product of vectors
$\text{u}={u}_{1}\text{i}+{u}_{2}\text{j}+{u}_{3}\text{k},$$\text{v}={v}_{1}\text{i}+{v}_{2}\text{j}+{v}_{3}\text{k},$ and
$\text{w}={w}_{1}\text{i}+{w}_{2}\text{j}+{w}_{3}\text{k}$ is the determinant of the
$3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}3$ matrix formed by the components of the vectors:
Let
$\text{u}=\u27e81,3,5\u27e9,\text{v}=\u27e82,\mathrm{-1},0\u27e9\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}=\u27e8\mathrm{-3},0,\mathrm{-1}\u27e9.$ Calculate the triple scalar product
$\text{u}\xb7\left(\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{w}\right).$
Calculate the triple scalar product
$\text{a}\xb7\left(\text{b}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{c}\right),$ where
$\text{a}=\u27e82,\mathrm{-4},1\u27e9,$$\text{b}=\u27e80,3,\mathrm{-1}\u27e9,$ and
$\text{c}=\u27e85,\mathrm{-3},3\u27e9.$
When we create a matrix from three vectors, we must be careful about the order in which we list the vectors. If we list them in a matrix in one order and then rearrange the rows, the absolute value of the determinant remains unchanged. However, each time two rows switch places, the determinant changes sign:
Rearranging vectors in the triple products is equivalent to reordering the rows in the matrix of the determinant. Let
$\text{u}={u}_{1}\text{i}+{u}_{2}\text{j}+{u}_{3}\text{k},$$\text{v}={v}_{1}\text{i}+{v}_{2}\text{j}+{v}_{3}\text{k},$ and
$\text{w}={w}_{1}\text{i}+{w}_{2}\text{j}+{w}_{3}\text{k}.$ Applying
[link] , we have
We can obtain the determinant for calculating
$\text{u}\xb7\left(\text{w}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}\right)$ by switching the bottom two rows of
$\text{u}\xb7\left(\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{w}\right).$ Therefore,
$\text{u}\xb7\left(\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{w}\right)=\text{\u2212}\text{u}\xb7\left(\text{w}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}\right).$
Following this reasoning and exploring the different ways we can interchange variables in the triple scalar product lead to the following identities:
Let
$\text{u}$ and
$\text{v}$ be two vectors in standard position. If
$\text{u}$ and
$\text{v}$ are not scalar multiples of each other, then these vectors form adjacent sides of a parallelogram. We saw in
[link] that the area of this parallelogram is
$\Vert \text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}\Vert .$ Now suppose we add a third vector
$\text{w}$ that does not lie in the same plane as
$\text{u}$ and
$\text{v}$ but still shares the same initial point. Then these vectors form three edges of a
parallelepiped , a three-dimensional prism with six faces that are each parallelograms, as shown in
[link] . The volume of this prism is the product of the figure’s height and the area of its base. The triple scalar product of
$\text{u},\text{v},$ and
$\text{w}$ provides a simple method for calculating the volume of the parallelepiped defined by these vectors.
Volume of a parallelepiped
The volume of a parallelepiped with adjacent edges given by the vectors
$\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}$ is the absolute value of the triple scalar product:
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?