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Mathematics-Online test:

Vector Calculus, Test 3

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Problem 1:
Solve the following linear system of equations:

\begin{displaymath} \begin{array}{rcrcrcc} 3x_1 & + & 2x_2 & + & 2x_3 & = & 2... ... & = & 3\\ -x_1 & + & 3x_2 & + & 4x_3 & = & 1 \end{array} \end{displaymath}

Solution:

$ x_1=$ ,      $ x_2=$ ,      $ x_3=$

(Specify the solution rounded on three decimal places)

Problem 2:
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 3:
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 4:

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 5:
Given the points $ P=(0,3,-2)$ , $ Q=(3,7,-1)$ and $ R=(1,-3,-1)$ in $ \mathbb{R}^3$ . Let $ g_1$ be the line through $ P$ and $ Q$ , and let $ g_2$ be the line through $ R$ with direction

$\displaystyle \vec{v}= \begin{pmatrix}1 \\ 0 \\ 1 \end{pmatrix} \; . $

Find the distance of $ P$ from $ g_2$ , and the distance between $ g_1$ and $ g_2$ .

Solution:

Distance of $ P$ from $ g_2$ : .

Distance between $ g_1$ and $ g_2$ : .

Problem 6:
Given the following plane in $ \mathbb{R}^3$

$\displaystyle E: \quad \vec{x}=\begin{pmatrix}-1 \\ 1 \\ 1 \end{pmatrix}+ \alp... ...\ 2 \\ 0 \end{pmatrix} + \beta \begin{pmatrix}1 \\ 0 \\ 1 \end{pmatrix} \; . $

a)
Let $ F$ be the plane through point $ A=(-4,2,2)$ , parallel to $ E$ . Find the equation describing the plane $ E$ .

b)
Which point $ B$ on plane $ E$ has minimal distance from point $ A$ . What is the minimal distance?

c)
Show that point $ C=(-3,0,4)$ lies in the plane $ F$ , and that the points $ A,B,C$ form an equilateral triangle. Find the lenghts of the sides, all interior angles, and the area of the triangle.

Solution:

a)
Complete the missing coefficients of the equation of $ F$ : $ 2x+$ $ y+$ $ z=$ .
b)
Point $ B=\Big($ , , $ \Big)$ , distance: .
c)
squared sides: $ \big\vert\overrightarrow{AB}\big\vert^2=$ , $ \big\vert\overrightarrow{BC}\big\vert^2=$ , $ \big\vert\overrightarrow{CA}\big\vert^2=$ .
$ \sphericalangle (ABC)=\pi/$ , $ \sphericalangle (BCA)=\pi/$ , $ \sphericalangle (CAB)=\pi/$ .
Area of triangle: /     (given as completely reduced fraction).
Problem 7:
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 1/11/2007