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
$9.$
Calculate price elasticity of demand and comment on the shape of the demand curve of a good ,when
its price rises by 20 percentage, quantity demanded falls from 150 units to 120 units.
5 %fall in price of good x leads to a 10 % rise in its quantity demanded. A 20 % rise in price of good y
leads to do a
10 % fall in its quantity demanded. calculate price elasticity of demand of good x and good y. Out of the
two goods which one is more elastic.
consider two goods X and Y. When the price of Y changes from 10 to 20. The quantity demanded of X changes from 40 to 35. Calculate cross elasticity of demand for X.
Sosna
sorry it the mistake answer it is question
Sosna
consider two goods X and Y. When the price of Y changes from 10 to 20. The quantity demanded of X changes from 40 to 35. Calculate cross elasticity of demand for X.
Sosna
The formula for calculation income elasticity of demand is the percent change in quantity demanded divided by the percent change in income.
n a perfectly competitive market, price equals marginal cost and firms earn an economic profit of zero. In a monopoly, the price is set above marginal cost and the firm earns a positive economic profit. Perfect competition produces an equilibrium in which the price and quantity of a good is economic
because monopoly have no competitor on the market and they are price makers,therefore,they can easily increase the princes and produce small quantity of goods but still consumers will still buy....
macroeconomics,microeconomics,positive economics and negative economics
Gladys
what are the factors of production
Gladys
process of production
Mutia
Basically factors of production are four (4) namely:
1. Entrepreneur
2. Capital
3. Labour and;
4. Land
but there has been a new argument to include an addition one to the the numbers to 5 which is "Technology"
Elisha
what is land as a factor of production
Gladys
what is Economic
Abu
economics is how individuals bussiness and governments make the best decisions to get what they want and how these choices interact in the market
Nandisha
Economics as a social science, which studies human behaviour as a relationship between ends and scarce means, which have alternative uses.
Yhaar
Economics is a science which study human behaviour as a relationship between ends and scarce means
John
Economics is a social sciences which studies human behavior as a relationship between ends and scarce mean, which have alternative uses.....
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
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?