Points and Vectors

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.
$C_1=[2,0],\ C_2=\left[-\frac{50}3,0\right]$
$C_1=[1,0],\ C_2=\left[-\frac{47}3,0\right]$
$C_1=[2,0],\ C_2=\left[-\frac{47}3,0\right]$
$C_1=[1,0],\ C_2=\left[-\frac{50}3,0\right]$

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.
$\vec{v}=(-6,6,-6)$
$\vec{v}=(6,-6,6)$
$\vec{v}=(-7,7,-7)$
$\vec{v}=(7,-7,7)$