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Mathematics-Online test:

Test 2 Vector Calculus

This test contains problems (P) with different versions (V).

Shown:  P1 V2   P2 V-   P3 V-   P4 V-   P5 V- 
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Problem 1:
Given vectors

$\displaystyle \vec{a}=\left(\begin{array}{c} 0 \\ 1 \\ 2 \end{array}\right),\qu... ...right),\quad \vec{c}=\left(\begin{array}{c} 2 \\ 3 \\ 4 \end{array}\right)\,. $

Find

$\displaystyle {\bf a)} \quad [\vec{a},\vec{b},\vec{c}\,]\,, \qquad {\bf b)} \qu... ...,, \qquad {\bf c)} \quad (\vec{a}\times\vec{c})\cdot (\vec{b}\times\vec{c})\,. $

Solution:

a)      $ [\vec{a},\vec{b},\vec{c}\,] = $

b)
$ (\vec{a}\times \vec{b})\times \vec{c}= \left(\rule{0pt}{6ex}\right.$
$ \left.\rule{0pt}{6ex}\right)$

c) $ \quad (\vec{a}\times\vec{c})\cdot (\vec{b}\times\vec{c})$ =

Problem 2:

Consider the triangle with vertices

$\displaystyle A=(2,-1,2),\quad B=(-1,5,-1),\quad C=(0,1,2).$

Compute the length of all sides, the angle $ \sphericalangle(\overrightarrow{AB},\overrightarrow{AC})$ and the aera of the triangle.

Antwort:
$ \big\vert\overrightarrow{AB}\big\vert^2=$ ,     $ \big\vert\overrightarrow{BC}\big\vert^2=$ ,     $ \big\vert\overrightarrow{AC}\big\vert^2=$ ,     $ \sphericalangle(\overrightarrow{AB},\overrightarrow{AC})=\pi/$ .
Square of the area of the triangle: .

Problem 3:
Given point $ P=(5,1,-3)$ , line

$\displaystyle g: \vec{x}=t\left(\begin{array}{c}3\\ 2\\ 1\end{array}\right), \qquad t\in\mathbb{R}, $

and plane $ E: 2x_1 -x_2 +2x_3 = 3$ . Find the projection of $ P$ onto $ g$ , the distance of $ P$ from $ E$ , and the point of intersection of $ g$ and $ E$ .

Solution:
projection:(,,) $ ^\mathrm{t}$
distance:
intersection: (,,)

Problem 4:
Consider a parallelogram $ ABCD$. Denote by $ M$ the centre of line segment $ \overline{BC}$ and let $ S$ denote the point of intersection of the line segment $ \overline{AM}$ and the diagonal $ \overline{BD}$.

a)
At what ratio does $ S$ partition the diagonal $ \overline{BD}$?
b)
What is the ratio of the area of the parallelogram $ ABCD$ to the area of the triangle $ BMS$?

\includegraphics[width=.6\linewidth]{a1_bild.eps}

Solution:

a)
The ratio $ \overline{BS} : \overline{SD}$ is : .
b)
The ratio area $ BMS$ : area $ ABCD$ is : .

Problem 5:
Let $ g$ be the line given by

$\displaystyle \vec{x}= \left( \begin{array}{c} 2 \\ -2 \\ -1 \end{array} \right) + t \left( \begin{array}{c} -1 \\ 0 \\ 1 \end{array} \right) $

and let $ P$ be the point $ (2,-1,1)$ .
a)
Calculate the distance between $ P$ and $ g$ .
b)
Determine the defining equation of the plane $ E$ spanned by the points $ A=(0,1,1)$ , $ B=(1,2,2)$ and $ C=(1,0,1)$ .
c)
Let $ F$ be the plane spanned by $ g$ and $ P$ . At which angle do $ E$ and $ F$ intersect each other?
d)
Calculate the volume of the parallelepiped spanned by $ A$ , $ B$ , $ C$ and $ D=(-3,0,1)$ .

Solution:

a)
Squared distance: .
b)
$ E$ : $ x+$ $ y+$ $ z=$ .
c)
Angel of intersection: $ \pi$ /.
d)
Volume: .

   
() automatically generated 6/16/2006