9000106802
Level:
A
Find the normal vector of the line passing through the point
\(A = [3;-1]\) and
\(B = [2;2]\).
\((3;1)\)
\((-1;3)\)
\((1;-3)\)
\((1;3)\)
Plane geometry
9000106803
Level:
A
Find the vector in the direction of the following line.
\[
p\colon y = \frac{2}
{3}x - 3
\]
\((3;2)\)
\(\left (\frac{2}
{3};-1\right )\)
\((3;-1)\)
\(\left (\frac{2}
{3};1\right )\)
9000106804
Level:
A
Find the normal vector of the following line.
\[
p\colon \begin{aligned}[t] x =&1 - 6t, &
\\y =& - 2 + 3t;\ t\in \mathbb{R}
\\ \end{aligned}
\]
\((1;2)\)
\((-6;3)\)
\((1;-2)\)
\((2;1)\)
9000107501
Level:
A
In the following list identify a line which is perpendicular to the line
\( 3x - 2y + 11 = 0\).
\(\begin{aligned}[t] x& = 3t, &
\\y & = 1 - 2t;\ t\in \mathbb{R}
\\ \end{aligned}\)
\(\begin{aligned}[t] x& = 1 + 2t, &
\\y & = 2 - 3t;\ t\in \mathbb{R}
\\ \end{aligned}\)
\(\begin{aligned}[t] x& = 2 - t, &
\\y & = 3 + t;\ t\in \mathbb{R}
\\ \end{aligned}\)
\(\begin{aligned}[t] x& = 2 + 3t, &
\\y & = 1 + 2t;\ t\in \mathbb{R}
\\ \end{aligned}\)
9000107502
Level:
A
In the following list identify a line which is perpendicular to the line
\(q\).
\[
\begin{aligned}q\colon x& = 5 - t,&
\\y & = 3t;\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(p\colon x - 3y - 7 = 0\)
\(p\colon - x - 3y + 11 = 0\)
\(p\colon 3x - y = 0\)
\(p\colon y = 5\)
9000107503
Level:
A
Among the lines in the following list (slope-intercept form) identify a line perpendicular
to the line
\[
y = \frac{3}{4}x + 1.
\]
\(y = -\frac{4}
{3}x - 2\)
\(y = -\frac{3}
{4}x - 1\)
\(y = \frac{4}
{3}x - 5\)
\(y = 3\)
9000107510
Level:
A
In the following list identify a line parallel to the line
\(q\).
\[
\begin{aligned}q\colon x& = t, &
\\y & = 1 + 5t;\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(p\colon - 5x + y - 13 = 0\)
\(p\colon x + 5y - 1 = 0\)
\(p\colon x - 5 = 0\)
\(p\colon 10x + 2y - 1 = 0\)
9000151310
Level:
A
Find the value of the real parameter
\(a\) which ensures, that
the following two lines \(p\)
and \(q\)
are perpendicular.
\[
p\colon ax + y - 4 = 0,\qquad q\colon x + 2y + 4 = 0.
\]
\(- 2\)
\(2\)
\(1\)
\(- 1\)
1003090802
Level:
B
Find the distance between parallel lines \( p \) and \( q \), if they are given by their general form equations, where \( p \) is \( 2x-4y+5=0 \) and \( q \) is \( x-2y+3=0 \).
\( \frac{\sqrt5}{10} \)
\( \frac{11\sqrt5}{10} \)
\( \frac{3}{2\sqrt5} \)
\( \frac{3\sqrt5}{10} \)
1003090803
Level:
B
Find the distance between parallel lines \( p \) and \( q \), if they are given by slope-intercept form equations, where \( p \) is \( y=-3x+5 \) and \( q \) is \( y=-3x-1 \).
\( \frac{3\sqrt{10}}5 \)
\( \frac{2\sqrt{10}}5 \)
\( \frac{4\sqrt{10}}5 \)
\( \frac{\sqrt{10}}5 \)