Then, by property i.,
$0\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{u}=0$ as well. Remember that the dot product of a vector and the zero vector is the
scalar$0,$ whereas the cross product of a vector with the zero vector is the
vector$0.$
Property
$\text{vi}.$ looks like the associative property, but note the change in operations:
Use the cross product properties to calculate
$\left(2\text{i}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}3\text{j}\right)\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{j}.$
Use the properties of the cross product to calculate
$\left(\text{i}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{k}\right)\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\left(\text{k}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{j}\right).$
So far in this section, we have been concerned with the direction of the vector
$\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v},$ but we have not discussed its magnitude. It turns out there is a simple expression for the magnitude of
$\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}$ involving the magnitudes of
$\text{u}$ and
$\text{v},$ and the sine of the angle between them.
Magnitude of the cross product
Let
$\text{u}$ and
$\text{v}$ be vectors, and let
$\theta $ be the angle between them. Then,
$\Vert \text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}\Vert =\Vert \text{u}\Vert \xb7\Vert \text{v}\Vert \xb7\text{sin}\phantom{\rule{0.2em}{0ex}}\theta .$
Proof
Let
$\text{u}=\u27e8{u}_{1},{u}_{2},{u}_{3}\u27e9$ and
$\text{v}=\u27e8{v}_{1},{v}_{2},{v}_{3}\u27e9$ be vectors, and let
$\theta $ denote the angle between them. Then
Taking square roots and noting that
$\sqrt{{\text{sin}}^{2}\theta}=\text{sin}\phantom{\rule{0.2em}{0ex}}\theta $ for
$0\le \theta \le 180\text{\xb0},$ we have the desired result:
This definition of the cross product allows us to visualize or interpret the product geometrically. It is clear, for example, that the cross product is defined only for vectors in three dimensions, not for vectors in two dimensions. In two dimensions, it is impossible to generate a vector simultaneously orthogonal to two nonparallel vectors.
Calculating the cross product
Use
[link] to find the magnitude of the cross product of
$\text{u}=\u27e80,4,0\u27e9$ and
$\text{v}=\u27e80,0,\mathrm{-3}\u27e9.$
Use
[link] to find the magnitude of
$\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v},$ where
$\text{u}=\u27e8\mathrm{-8},0,0\u27e9$ and
$\text{v}=\u27e80,2,0\u27e9.$
Using
[link] to find the cross product of two vectors is straightforward, and it presents the cross product in the useful component form. The formula, however, is complicated and difficult to remember. Fortunately, we have an alternative. We can calculate the cross product of two vectors using
determinant notation.
A
$2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}2$ determinant is defined by
A
$3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}3$ determinant is defined in terms of
$2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}2$ determinants as follows:
[link] is referred to as the
expansion of the determinant along the first row . Notice that the multipliers of each of the
$2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}2$ determinants on the right side of this expression are the entries in the first row of the
$3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}3$ determinant. Furthermore, each of the
$2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}2$ determinants contains the entries from the
$3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}3$ determinant that would remain if you crossed out the row and column containing the multiplier. Thus, for the first term on the right,
${a}_{1}$ is the multiplier, and the
$2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}2$ determinant contains the entries that remain if you cross out the first row and first column of the
$3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}3$ determinant. Similarly, for the second term, the multiplier is
${a}_{2},$ and the
$2\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}2$ determinant contains the entries that remain if you cross out the first row and second column of the
$3\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}3$ determinant. Notice, however, that the coefficient of the second term is negative. The third term can be calculated in similar fashion.
Questions & Answers
where we get a research paper on Nano chemistry....?
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?