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Mathematics-Online test:

Test 1 Vector Calculus


Problem 1:
Let $ P$ be the point with coordinates $ (\sqrt{6},-\sqrt{6},2)$ with respect to the Cartesian coordinate system spanned by the canonical unit vectors of $ \mathbb{R}^3$ . Find the spherical and the cylindrical coordinates of $ P$ . Which Cartesian coordinates does point $ P$ have, if we rotate the coordinate system about the $ x$ -axis by $ \pi/3$ .

Solution:
Spherical coordinates of $ P$ : $ r=$ ,      $ \varphi=$ $ \pi/$ ,      $ \vartheta=\pi/$
Cylindrical coordinates of $ P$ : $ \rho=\big($ $ \big)^{1/2}$ ,     $ \varphi=$ $ \pi/$ ,     $ z=$
Cartesian coordinates after rotation (Results should be rounded to 4 decimal digits): $ P'=\Big($ ,,$ \Big)$


Problem 2:
Let $ \vec{a}, \vec{b}, \vec{c}$ be vectors in $ \mathbb{R}^3$ . Decide whether the following statements are true or false.
a)
$ \vec{a} \times \vec{b} = \vec{0}$ implies that at least one of the two vectors $ \vec{a}, \vec{b}$ is the zero vector.
b)
The following holds: $ (\vec{a}-\vec{b})\times
(\vec{a}+\vec{b})=2(\vec{a}\times \vec{b})$ .
c)
Every orthonormal basis form a right-handed system.
d)
The following holds: $ \big\vert\vert\vec{a}-\vec{c}\vert-\vert\vec{b}-\vec{c}\vert\big\vert\leq
\vert\vec{a}-\vec{b}\vert$ .
e)
If vector $ \vec{a}$ is a multiple of vector $ \vec{b}$ , then $ [\vec{a}, \vec{b}, \vec{c}]=0$ .

Solution:
a) true      false
b) true false
c) true false
d) true false
e) true false


Problem 3:
Show that the vectors

$\displaystyle \vec{u}= \begin{pmatrix}6 \\ -3 \\ 2 \end{pmatrix}, \quad
\vec{v...
...\\ 3 \end{pmatrix}, \quad
\vec{w}= \begin{pmatrix}3 \\ 2 \\ -6 \end{pmatrix}
$

are mutually orthogonal. Do they form a left-handed or a right-handed system?

Keine Angabe ,     left-handed ,      right-handed .

Calculate the magnitudes $ \vert\vec{u}\vert$, $ \vert\vec{v}\vert$, $ \vert\vec{w}\vert$ and find $ \alpha,\beta,\gamma\in \mathbb{R}$ so that

$\displaystyle \frac{\alpha}{\vert\vec{u}\vert}\,\vec{u} +
\frac{\beta}{\vert\v...
...vert\vec{w}\vert}\,\vec{w} =
\begin{pmatrix}14 \\ -7 \\ 0 \end{pmatrix} \; .
$

Solution:

Magnitudes:
$ \vert\vec{u}\vert=$,      $ \vert\vec{v}\vert=$,      $ \vert\vec{w}\vert=$.

Parameter:
$ \alpha=$,     $ \beta=$,     $ \gamma=$.


Problem 4:
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 5:
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).

   

(conceptual design by Joachim Wipper) automatically generated 8/11/2017