Consider points
$A\left(\alpha ,0,0\right),B\left(0,\beta ,0\right),$ and
$C\left(0,0,\gamma \right),$ with
$\alpha ,$$\beta ,$ and
$\gamma $ positive real numbers.
Determine the volume of the parallelepiped with adjacent sides
$\overrightarrow{OA},$$\overrightarrow{OB},$ and
$\overrightarrow{OC}.$
Find the volume of the tetrahedron with vertices
$O,A,B,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}C.$ (
Hint : The volume of the tetrahedron is
$1\text{/}6$ of the volume of the parallelepiped.)
Find the distance from the origin to the plane determined by
$A,B,\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}C.$ Sketch the parallelepiped and tetrahedron.
Let
$\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}$ be three-dimensional vectors and
$c$ be a real number. Prove the following properties of the cross product.
Show that vectors
$\text{u}=\u27e81,0,\mathrm{-8}\u27e9,$$\text{v}=\u27e80,1,6\u27e9,$ and
$\text{w}=\u27e8\mathrm{-1},9,3\u27e9$ satisfy the following properties of the cross product.
Nonzero vectors
$\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}$ are said to be
linearly dependent if one of the vectors is a linear combination of the other two. For instance, there exist two nonzero real numbers
$\alpha $ and
$\beta $ such that
$\text{w}=\alpha \text{u}+\beta \text{v}.$ Otherwise, the vectors are called
linearly independent . Show that
$\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}$ are coplanar if and only if they are linear dependent.
Consider vectors
$\text{u}=\u27e81,4,\mathrm{-7}\u27e9,$$\text{v}=\u27e82,\mathrm{-1},4\u27e9,$$\text{w}=\u27e80,\mathrm{-9},18\u27e9,$ and
$\mathbf{\text{p}}=\u27e80,\mathrm{-9},17\u27e9.$
Show that
$\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}$ are coplanar by using their triple scalar product
Show that
$\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{w}$ are coplanar, using the definition that there exist two nonzero real numbers
$\alpha $ and
$\beta $ such that
$\text{w}=\alpha \text{u}+\beta \text{v}.$
Show that
$\text{u},\text{v},\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{p}$ are linearly independent—that is, none of the vectors is a linear combination of the other two.
Consider points
$A(0,0,2),$$B\left(1,0,2\right),$$C\left(1,1,2\right),$ and
$D\left(0,1,2\right).$ Are vectors
$\overrightarrow{AB},$$\overrightarrow{AC},$ and
$\overrightarrow{AD}$ linearly dependent (that is, one of the vectors is a linear combination of the other two)?
Yes,
$\overrightarrow{AD}=\alpha \overrightarrow{AB}+\beta \overrightarrow{AC},$ where
$\alpha =\mathrm{-1}$ and
$\beta =1.$
Show that vectors
$\text{i}+\text{j},$$\text{i}-\text{j},$ and
$\text{i}+\text{j}+\text{k}$ are linearly independent—that is, there exist two nonzero real numbers
$\alpha $ and
$\beta $ such that
$\text{i}+\text{j}+\text{k}=\alpha \left(\text{i}+\text{j}\right)+\beta \left(\text{i}-\text{j}\right).$
Let
$\text{u}=\u27e8{u}_{1},{u}_{2}\u27e9$ and
$\text{v}=\u27e8{v}_{1},{v}_{2}\u27e9$ be two-dimensional vectors. The cross product of vectors
$\text{u}$ and
$\text{v}$ is not defined. However, if the vectors are regarded as the three-dimensional vectors
$\tilde{\text{u}}=\u27e8{u}_{1},{u}_{2},0\u27e9$ and
$\tilde{\text{v}}=\u27e8{v}_{1},{v}_{2},0\u27e9,$ respectively, then, in this case, we can define the cross product of
$\tilde{\text{u}}$ and
$\tilde{\text{v}}.$ In particular, in determinant notation, the cross product of
$\tilde{\text{u}}$ and
$\tilde{\text{v}}$ is given by
Use this result to compute
$(\text{i}\text{cos}\phantom{\rule{0.2em}{0ex}}\theta +\text{j}\text{sin}\phantom{\rule{0.2em}{0ex}}\theta )\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}(\text{i}sin\theta -\text{j}cos\theta ),$ where
$\theta $ is a real number.
Consider
$\text{u}$ and
$\text{v}$ two three-dimensional vectors. If the magnitude of the cross product vector
$\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{v}$ is
$k$ times larger than the magnitude of vector
$\text{u},$ show that the magnitude of
$\text{v}$ is greater than or equal to
$k,$ where
$k$ is a natural number.
[T] Assume that the magnitudes of two nonzero vectors
$\text{u}$ and
$\text{v}$ are known. The function
$f(\theta )=\Vert \text{u}\Vert \Vert \text{v}\Vert \text{sin}\phantom{\rule{0.2em}{0ex}}\theta $ defines the magnitude of the cross product vector
$\text{u}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\mathbf{\text{v}},$ where
$\theta \in [0,\pi ]$ is the angle between
$\text{u}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{v}.$
Graph the function
$f.$
Find the absolute minimum and maximum of function
$f.$ Interpret the results.
If
$\Vert \text{u}\Vert =5$ and
$\Vert \text{v}\Vert =2,$ find the angle between
$\text{u}\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}\text{v}$ if the magnitude of their cross product vector is equal to
$9.$
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?