Technically, determinants are defined only in terms of arrays of real numbers. However, the determinant notation provides a useful mnemonic device for the cross product formula.
Rule: cross product calculated by a determinant
Let
$\text{u}=\u27e8{u}_{1},{u}_{2},{u}_{3}\u27e9$ and
$\text{v}=\u27e8{v}_{1},{v}_{2},{v}_{3}\u27e9$ be vectors. Then the cross product
$\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}$ is given by
Using determinant notation to find
$\text{p}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{q}$
Let
$\text{p}=\u27e8\mathrm{-1},2,5\u27e9$ and
$\text{q}=\u27e84,0,\mathrm{-3}\u27e9.$ Find
$\text{p}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{q}.$
We set up our determinant by putting the standard unit vectors across the first row, the components of
$\text{u}$ in the second row, and the components of
$\text{v}$ in the third row. Then, we have
Use determinant notation to find
$\text{a}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{b},$ where
$\text{a}=\u27e88,2,3\u27e9$ and
$\text{b}=\u27e8\mathrm{-1},0,4\u27e9.$
The cross product is very useful for several types of calculations, including finding a vector orthogonal to two given vectors, computing areas of triangles and parallelograms, and even determining the volume of the three-dimensional geometric shape made of parallelograms known as a
parallelepiped . The following examples illustrate these calculations.
Finding a unit vector orthogonal to two given vectors
Let
$\text{a}=\u27e85,2,\mathrm{-1}\u27e9$ and
$\text{b}=\u27e80,\mathrm{-1},4\u27e9.$ Find a unit vector orthogonal to both
$\text{a}$ and
$\text{b}.$
The cross product
$\text{a}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{b}$ is orthogonal to both vectors
$\text{a}$ and
$\text{b}.$ We can calculate it with a determinant:
Thus,
$\u27e8\frac{7}{\sqrt{474}},\frac{\mathrm{-20}}{\sqrt{474}},\frac{\mathrm{-5}}{\sqrt{474}}\u27e9$ is a unit vector orthogonal to
$\text{a}$ and
$\text{b}.$
To use the cross product for calculating areas, we state and prove the following theorem.
Area of a parallelogram
If we locate vectors
$\text{u}$ and
$\text{v}$ such that they form adjacent sides of a parallelogram, then the area of the parallelogram is given by
$\Vert \text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}\Vert $ (
[link] ).
Proof
We show that the magnitude of the cross product is equal to the base times height of the parallelogram.
Let
$P=\left(1,0,0\right),Q=\left(0,1,0\right),\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}R=\left(0,0,1\right)$ be the vertices of a triangle (
[link] ). Find its area.
We have
$\overrightarrow{PQ}=\u27e80-1,1-0,0-0\u27e9=\u27e8\mathrm{-1},1,0\u27e9$ and
$\overrightarrow{PR}=\u27e80-1,0-0,1-0\u27e9=\u27e8\mathrm{-1},0,1\u27e9.$ The area of the parallelogram with adjacent sides
$\overrightarrow{PQ}$ and
$\overrightarrow{PR}$ is given by
$\Vert \overrightarrow{PQ}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\overrightarrow{PR}\Vert \text{:}$
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?