9000106804
Level:
A
Find the normal vector of the following line.
\[
p\colon \begin{aligned}[t] x =&1 - 6t, &
\\y =& - 2 + 3t;\ t\in \mathbb{R}
\\ \end{aligned}
\]
\((1;2)\)
\((-6;3)\)
\((1;-2)\)
\((2;1)\)
Analytical plane geometry
9000107501
Level:
A
In the following list identify a line which is perpendicular to the line
\( 3x - 2y + 11 = 0\).
\(\begin{aligned}[t] x& = 3t, &
\\y & = 1 - 2t;\ t\in \mathbb{R}
\\ \end{aligned}\)
\(\begin{aligned}[t] x& = 1 + 2t, &
\\y & = 2 - 3t;\ t\in \mathbb{R}
\\ \end{aligned}\)
\(\begin{aligned}[t] x& = 2 - t, &
\\y & = 3 + t;\ t\in \mathbb{R}
\\ \end{aligned}\)
\(\begin{aligned}[t] x& = 2 + 3t, &
\\y & = 1 + 2t;\ t\in \mathbb{R}
\\ \end{aligned}\)
9000106803
Level:
A
Find the vector in the direction of the following line.
\[
p\colon y = \frac{2}
{3}x - 3
\]
\((3;2)\)
\(\left (\frac{2}
{3};-1\right )\)
\((3;-1)\)
\(\left (\frac{2}
{3};1\right )\)
9000107503
Level:
A
Among the lines in the following list (slope-intercept form) identify a line perpendicular
to the line
\[
y = \frac{3}{4}x + 1.
\]
\(y = -\frac{4}
{3}x - 2\)
\(y = -\frac{3}
{4}x - 1\)
\(y = \frac{4}
{3}x - 5\)
\(y = 3\)
9000106805
Level:
C
Given points \(A = [0;5]\),
\(B = [6;1]\),
\(C = [7;9]\),
find the direction vector of the line passing through the point
\(A\) and the midpoint of
the segment \(BC\) (i.e. the
median of the triangle \(ABC\)
through the vertex \(A\)).
\((1;0)\)
\((1;8)\)
\((1;9)\)
\((6.5;5)\)
9000107502
Level:
A
In the following list identify a line which is perpendicular to the line
\(q\).
\[
\begin{aligned}q\colon x& = 5 - t,&
\\y & = 3t;\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(p\colon x - 3y - 7 = 0\)
\(p\colon - x - 3y + 11 = 0\)
\(p\colon 3x - y = 0\)
\(p\colon y = 5\)
9000107504
Level:
B
Find the angle between the lines
\[
p\colon 2x - 3y + 1 = 0;\quad q\colon 3x + 2y - 3 = 0.
\]
\(90^{\circ }\)
\(60^{\circ }\)
\(0^{\circ }\)
\(30^{\circ }\)
9000107506
Level:
B
Find \(\cos \varphi \) where
\(\varphi \) is the angle
between the lines \(p\)
and \(q\).
\[
p\colon y = 2x - 11;\quad q\colon y = \frac{1}
{4}x
\]
\(\frac{6\sqrt{85}}
{85} \)
\(\frac{1}
{\sqrt{22}}\)
\(\frac{\sqrt{6}}
{85} \)
\(\frac{\sqrt{17}}
{30} \)
9000107505
Level:
B
Find \(\cos \varphi \) where
\(\varphi \) is the angle
between the lines \(p\)
and \(q\).
\[
\begin{aligned}[t] p\colon x& = 1 + 4t, &
\\y & = 3 - 3t;\ t\in \mathbb{R};
\\ \end{aligned} \quad q\colon x + y - 3 = 0
\]
\(\frac{7\sqrt{2}}
{10} \)
\(- \frac{7}
{5\sqrt{2}}\)
\(\frac{\sqrt{2}}
{5} \)
\(\frac{\sqrt{2}}
{10} \)
9000107507
Level:
B
Find \(\mathop{\mathrm{tg}}\nolimits \varphi \) where
\(\varphi \) is the angle
between the lines \(p\)
and \(q\).
\[
\begin{aligned}[t] p\colon x& = 1 + t, &
\\y & = 3 + 2t;\ t\in \mathbb{R};
\\ \end{aligned}\quad q\colon y = 1
\]
\(2\)
\(\frac{1}
{2}\)
\(- 1\)
\(0\)