Notice what this means for the direction of
$\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{u}.$ If we apply the right-hand rule to
$\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{u},$ we start with our fingers pointed in the direction of
$\text{v},$ then curl our fingers toward the vector
$\text{u}.$ In this case, the thumb points in the opposite direction of
$\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}.$ (Try it!)
Anticommutativity of the cross product
Let
$\text{u}=\u27e80,2,1\u27e9$ and
$\text{v}=\u27e83,\mathrm{-1},0\u27e9.$ Calculate
$\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}$ and
$\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{u}$ and graph them.
We see that, in this case,
$\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}=\text{\u2212}\left(\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{u}\right)$ (
[link] ). We prove this in general later in this section.
Suppose vectors
$\text{u}$ and
$\text{v}$ lie in the
xy -plane (the
z -component of each vector is zero). Now suppose the
x - and
y -components of
$\text{u}$ and the
y -component of
$\text{v}$ are all positive, whereas the
x -component of
$\text{v}$ is negative. Assuming the coordinate axes are oriented in the usual positions, in which direction does
$\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}$ point?
The cross products of the standard unit vectors
$\text{i},\text{j},$ and
$\text{k}$ can be useful for simplifying some calculations, so let’s consider these cross products. A straightforward application of the definition shows that
(The cross product of two vectors is a vector, so each of these products results in the zero vector, not the scalar
$0.)$ It’s up to you to verify the calculations on your own.
Furthermore, because the cross product of two vectors is orthogonal to each of these vectors, we know that the cross product of
$\text{i}$ and
$\text{j}$ is parallel to
$\text{k}.$ Similarly, the vector product of
$\text{i}$ and
$\text{k}$ is parallel to
$\text{j},$ and the vector product of
$\text{j}$ and
$\text{k}$ is parallel to
$\text{i}.$ We can use the right-hand rule to determine the direction of each product. Then we have
We know that
$\text{j}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{k}=\text{i}.$ Therefore,
$\text{i}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\left(\text{j}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{k}\right)=\text{i}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{i}=0.$
As we have seen, the dot product is often called the
scalar product because it results in a scalar. The cross product results in a vector, so it is sometimes called the
vector product . These operations are both versions of vector multiplication, but they have very different properties and applications. Let’s explore some properties of the cross product. We prove only a few of them. Proofs of the other properties are left as exercises.
Properties of the cross product
Let
$\text{u},\text{v},$ and
$\text{w}$ be vectors in space, and let
$c$ be a scalar.
For property
$\text{i}.,$ we want to show
$\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}=\text{\u2212}\left(\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{u}\right).$ We have
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?
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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?