1103040204
Level:
C
We are given the points $A = [1;2;1]$, $B = [7;3;0]$, $C = [-1;5;2]$ and $D = [1;0;6]$.
Find the volume of the triangular prism $ABCDEF$ shown in the picture.
$V=54$
$V=108$
$V=36$
$V=56$
Points and vectors
1103040203
Level:
C
We are given the points $A = [1 ; -2 ; 3]$, $B = [1 ; -2 ; -1]$, $C = [6 ; 10 ; -1]$ and $D = [4 ; -2 ; 3]$.
Find the volume of the tetrahedron $ABCD$ shown in the picture.
$V=24$
$V=48$
$V=72$
$V=16$
1103040202
Level:
C
We are given the points $A = [1 ; -2 ; -3]$, $B = [4 ; 1 ; -1]$, $D = [-3 ; 3 ; 1]$ and $E = [2 ; 0 ; 5]$ (see the picture).
Find the volume of the pyramid $ABCDE$ with the parallelogram base $ABCD$ and the apex $E$.
$V=\frac{178}3$
$V=\frac{89}3$
$V=178$
$V=89$
1003040201
Level:
C
We are given the vectors $\vec{a}=(-1; 2;3)$, $\vec{b}=(3; 1; -2)$ and $\vec{c}=(1; 2;-1)$. Find the coordinates of a vector $\vec{v}$, such that $\vec{v}$ is perpendicular to both vectors $\vec{a}$ and $\vec{b}$, while $\vec{v}\cdot\vec{c}=12$ holds.
$\vec{v}=(-6;6;-6)$
$\vec{v}=(6;-6;6)$
$\vec{v}=(-7;7;-7)$
$\vec{v}=(7;-7;7)$
1103030505
Level:
B
The vectors \( \overrightarrow{u} \) and \( \overrightarrow{v} \) are given by the figure. Find cosine of the angle \( \varphi \) between \( \overrightarrow{u} \) and \( \overrightarrow{v} \).
Help: Use the dot product of the given vectors.
\( \cos\varphi=-\frac9{17} \)
\( \cos\varphi=\frac9{17} \)
\( \cos\varphi=\frac{\sqrt{17}}{2\sqrt{13}} \)
\( \cos\varphi=-\frac{\sqrt{17}}{2\sqrt{13}} \)
1103030504
Level:
B
The vectors \( \overrightarrow{u} \) and \( \overrightarrow{v} \) are given by the figure. Find cosine of the angle \(\varphi \) between \( \overrightarrow{u} \) and \( \overrightarrow{v} \).
Help: Use the dot product of the given vectors.
\( \cos\varphi=\frac{13\sqrt{10}}{50} \)
\( \cos\varphi=\frac{970}{50} \)
\( \cos\varphi=\frac{3\sqrt{10}}{10} \)
\( \cos\varphi=\frac{\sqrt{10}}{5} \)
1103030503
Level:
B
Find the coordinates of the vectors \( \overrightarrow{u} \) and \( \overrightarrow{v} \) given by the picture and evaluate their dot product.
\( \overrightarrow{u}=(-8;-7;9);\ \ \overrightarrow{v} =(8;7;9);\ \ \overrightarrow{u}\cdot\overrightarrow{v} = -32 \)
\( \overrightarrow{u}=(-8;-7;9);\ \ \overrightarrow{v} =(8;7;9);\ \ \overrightarrow{u}\cdot\overrightarrow{v} = 0 \)
\( \overrightarrow{u}=(-8;-7;9);\ \ \overrightarrow{v} =(8;7;9);\ \ \overrightarrow{u}\cdot\overrightarrow{v} = (-64;-49;81) \)
\( \overrightarrow{u}=(8;7;-9);\ \ \overrightarrow{v} =(-8;-7;-9);\ \ \overrightarrow{u}\cdot\overrightarrow{v} = (-64;-49;81) \)
1103030502
Level:
B
Find the coordinates of the vectors \( \overrightarrow{u} \) and \( \overrightarrow{v} \) given by the picture and evaluate their dot product.
\( \overrightarrow{u}=(-3;6);\ \ \overrightarrow{v} =(-9;-6);\ \ \overrightarrow{u}\cdot\overrightarrow{v} = -9 \)
\( \overrightarrow{u}=(3;-6);\ \ \overrightarrow{v} =(9;6);\ \ \overrightarrow{u}\cdot\overrightarrow{v} = -9 \)
\( \overrightarrow{u}=(-3;6);\ \ \overrightarrow{v} =(-9;-6);\ \ \overrightarrow{u}\cdot\overrightarrow{v} = 9 \)
\( \overrightarrow{u}=(3;-6);\ \ \overrightarrow{v} =(9;6);\ \ \overrightarrow{u}\cdot\overrightarrow{v} = 0 \)
1103030501
Level:
B
The vectors \( \overrightarrow{u} \), \( \overrightarrow{v}\), \( \overrightarrow{w} \), \( \overrightarrow{z} \) are indicated in a cube shown in the figure. The cube edge length is \( 1 \). Find the dot products of:
\[ \overrightarrow{v}\cdot\overrightarrow{z}\text{ ,}\ \ \overrightarrow{u}\cdot\overrightarrow{v} \text{ ,}\ \ \overrightarrow{w}\cdot\overrightarrow{u}\]
\( \overrightarrow{v}\cdot\overrightarrow{z}=1 \), \( \overrightarrow{u}\cdot\overrightarrow{v}=0 \), \( \overrightarrow{w}\cdot\overrightarrow{u}=1 \)
\( \overrightarrow{v}\cdot\overrightarrow{z}=\frac{\sqrt2}2 \), \( \overrightarrow{u}\cdot\overrightarrow{v}=1 \), \( \overrightarrow{w}\cdot\overrightarrow{u}=\sqrt3 \)
\( \overrightarrow{v}\cdot\overrightarrow{z}=\sqrt2 \), \( \overrightarrow{u}\cdot\overrightarrow{v}=0 \), \( \overrightarrow{w}\cdot\overrightarrow{u}=1 \)
\( \overrightarrow{v}\cdot\overrightarrow{z}=1 \), \( \overrightarrow{u}\cdot\overrightarrow{v}=1 \), \( \overrightarrow{w}\cdot\overrightarrow{u}=\sqrt3 \)
1003020807
Level:
A
Find the set of all values of a parameter \( a\in\mathbb{R} \) such that the distance of the points \( [a;3;-2] \) and \( [1;1;4] \) equals \( 7 \).
\( \{4;-2\} \)
\( \{4;2\} \)
\( \emptyset \)
\( \{0\} \)