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
$\Vert \overrightarrow{PQ}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\overrightarrow{PR}\Vert \text{:}$
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest.
Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.?
How this robot is carried to required site of body cell.?
what will be the carrier material and how can be detected that correct delivery of drug is done
Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?