9000106803
Level:
A
Find the vector in the direction of the following line.
\[
p\colon y = \frac{2}
{3}x - 3
\]
\((3,2)\)
\(\left (\frac{2}
{3},-1\right )\)
\((3,-1)\)
\(\left (\frac{2}
{3},1\right )\)
Analytical Plane Geometry
9000106804
Level:
A
Find the normal vector of the following line.
\[
p\colon \begin{aligned}[t] x =&1 - 6t, &
\\y =& - 2 + 3t,\ t\in \mathbb{R}
\\ \end{aligned}
\]
\((1,2)\)
\((-6,3)\)
\((1,-2)\)
\((2,1)\)
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 \(A\)).
\((1,0)\)
\((1,8)\)
\((1,9)\)
\((6.5,5)\)
9000106806
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 perpendicular
to the segment \(BC\)
(i.e. the line which contains the altitude of the triangle
\(ABC\) through
the point \(A\)).
\((8,-1)\)
\((1,8)\)
\((1,9)\)
\((-9,1)\)
9000106807
Level:
C
Consider the points \(A = [0,5]\),
\(B = [6,1]\),
\(C = [7,9]\) and the
triangle \(ABC\).
Find the direction vector of the line which is the perpendicular bisector of the side
\(b\)
(i.e. the line through the midpoint of the side
\(AC\) which is perpendicular
to the segment \(AC\)).
\((4,-7)\)
\((7,4)\)
\((7,9)\)
\((7,-9)\)
9000106808
Level:
C
Consider the points \(A = [0,5]\),
\(B = [6,1]\),
\(C = [7,9]\) and the
triangle \(ABC\).
Find the direction vector of the line which is the bisector of the angle
\(ACB\)
(i.e. the line which splits the internal angle at the point
\(C\) into
two angles with equal measures).
\((2,3)\)
\((6,-4)\)
\((7,9)\)
\((7,8)\)
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
\(q\).
\[
\begin{aligned}q\colon x& = 5 - t,&
\\y & = 3t,\ t\in \mathbb{R}
\\ \end{aligned}
\]
\(p\colon x - 3y - 7 = 0\)
\(p\colon - x - 3y + 11 = 0\)
\(p\colon 3x - y = 0\)
\(p\colon y = 5\)
9000107504
Level:
B
Find the angle between the lines
\[
p\colon 2x - 3y + 1 = 0,\quad q\colon 3x + 2y - 3 = 0.
\]
\(90^{\circ }\)
\(60^{\circ }\)
\(0^{\circ }\)
\(30^{\circ }\)
9000107505
Level:
B
Find \(\cos \varphi \) where
\(\varphi \) is the angle
between the lines \(p\)
and \(q\).
\[
\begin{aligned}[t] p\colon x& = 1 + 4t, &
\\y & = 3 - 3t,\ t\in \mathbb{R},
\\ \end{aligned} \quad q\colon x + y - 3 = 0
\]
\(\frac{7\sqrt{2}}
{10} \)
\(- \frac{7}
{5\sqrt{2}}\)
\(\frac{\sqrt{2}}
{5} \)
\(\frac{\sqrt{2}}
{10} \)