Points and Vectors
1003040210
Level:
B
Given the points $A = [3;3;0]$ and $B = [0;3;3]$, specify all the points $C$ lying on the $y$-axis, such that $|\measuredangle ABC|=\frac{\pi}3$ holds.
$C_1=[0;0;0];\ C_2=[0;6;0]$
$C_1=[0;3;0];\ C_2=[0;9;0]$
$C_1=[0;-3;0];\ C_2=[0;3;0]$
$C_1=[0;-6;0];\ C_2=[0;6;0]$
1103040209
Level:
B
In the picture, there are indicated vectors $\vec{u}$ and $\vec{v}$ in three squares. Find the measure of an angle $\varphi$ between $\vec{u}$ and $\vec{v}$. Round $\varphi$ to the nearest degree.
Hint: Set up a coordinate system conveniently.
$\varphi\doteq 8^{\circ}$
$\varphi\doteq 9^{\circ}$
$\varphi\doteq 10^{\circ}$
$\varphi\doteq 11^{\circ}$
1103040208
Level:
C
We are given the points $A = [4;5;-1]$, $B = [-2;-1;2]$, $C = [-1;-3;0]$ and $D = [0;m;2]$. Find the missing coordinate of the point $D$ such that the point $D$ lies in the plane determined by the points $A$, $B$ and $C$.
Hint: Use a linear combination of vectors shown in the picture or use their mixed product.
$m=3$
$m=-3$
$m=1$
$m$ does not exist
1003040207
Level:
C
Given the points $A = [2;0;3]$ and $B = [-1;2;0]$, specify all the points $C$ lying on the $z$-axis, such that the area of the triangle $ABC$ is $2\sqrt2$.
Hint: Use a cross product of vectors.
$C_1=[0;0;1];\ C_2=\left[0;0;\frac{29}{13}\right]$
$C_1=[0;0;1];\ C_2=\left[0;0;-1\right]$
$C_1=[0;0;-1];\ C_2=\left[0;0;\frac{13}{29}\right]$
$C_1=[0;0;-1];\ C_2=\left[0;0;\frac{29}{13}\right]$
1103040206
Level:
C
Given the points $A = [1;5]$ and $B = [-4;2]$, specify all the points $C$ lying on the $x$-axis, such that the area of the triangle $ABC$ is $14$.
Hint: Use a cross product of vectors.
$C_1=[2;0];\ C_2=\left[-\frac{50}3;0\right]$
$C_1=[1;0];\ C_2=\left[-\frac{47}3;0\right]$
$C_1=[2;0];\ C_2=\left[-\frac{47}3;0\right]$
$C_1=[1;0];\ C_2=\left[-\frac{50}3;0\right]$
1003040205
Level:
C
We are given the vectors $\vec{a}=(1;-2;-2)$, $\vec{b}=(0;1;3)$ and $\vec{c}=(1;-1;0)$. Find the mixed product $(\vec{a}\times\vec{b})\cdot\vec{c}$.
$-1$
$(1;-2;-2)$
The mixed product is not defined.
$(-8;8;0)$
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.
$V=54$
$V=108$
$V=36$
$V=56$
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 $E$.
$V=\frac{178}3$
$V=\frac{89}3$
$V=178$
$V=89$