1003020802
Level:
A
\( S \) is the midpoint of \( AB \). \( A = [5; -7] \), \( S = [0; -9] \). Find the coordinates of \( B \).
\( B = [-5; -11] \)
\( B = [5; -18] \)
\( B = \left[-\frac52; -\frac{11}2\right] \)
\( B = [5; 11] \)
Points and vectors
1103020801
Level:
A
Find the coordinates of the midpoints of the line segments \( AB \), \( BC \), \( AC \). For coordinates of the points \( A \), \( B \) and \( C \), see the picture.
\( S_{AB}=\left[-\frac12;1 \right]\text{, }\ S_{BC}=[4;2 ]\text{, }\ S_{AC}=\left[\frac12; 4\right] \)
\( S_{AB}=\left[-\frac32;2 \right]\text{, }\ S_{BC}=[1;3 ]\text{, }\ S_{AC}=\left[\frac52; 4\right] \)
\( S_{AB}=\left[\frac12;1 \right]\text{, }\ S_{BC}=[4;2 ]\text{, }\ S_{AC}=\left[-\frac12; 4\right] \)
\( S_{AB}=\left[1;-\frac12 \right]\text{, }\ S_{BC}=[2;4 ]\text{, }\ S_{AC}=\left[4;\frac12\right] \)
1103020808
Level:
A
Let \( ABC \) be a triangle. In the picture, the midpoint of the side \( BC \) and the centroid of the triangle are indicated. Out of the following vector relations select the one that is not true.
\( \overrightarrow{ST}= \frac12 \overrightarrow{AT} \)
\( \overrightarrow{AT}= \frac23\overrightarrow{AS} \)
\( \overrightarrow{ST} = -\frac13\overrightarrow{AS} \)
\( \overrightarrow{SA}= -3\overrightarrow{TS} \)
1003030605
Level:
B
Let \( \overrightarrow{a}=(3;-5) \) and \( \overrightarrow{b}=(6;-10) \). Find all the vectors \( \overrightarrow{c} \) such that
\[ \overrightarrow{a}\cdot\overrightarrow{c}=11\ \text{ and }\ \overrightarrow{b}\cdot\overrightarrow{c}=22\text{ .} \]
\( \overrightarrow{c}=(2+5k;-1+3k);\ k\in\mathbb{R} \)
\( \overrightarrow{c}_1=(7;2);\ \overrightarrow{c}_2=(-7;-2) \)
\( \overrightarrow{c}=(2k;-k);\ k\in\mathbb{R} \)
\( \overrightarrow{c}_1=(2;-1);\ \overrightarrow{c}_2=(-2;1) \)
1003030604
Level:
B
Let \( \overrightarrow{a}=(2;- 3) \) and \( \overrightarrow{b}=(3;-2) \). Find all the vectors \( \overrightarrow{c} \) such that
\[ \overrightarrow{a}\cdot\overrightarrow{c}=8\ \text{ and }\ \overrightarrow{b}\cdot\overrightarrow{c}=27. \]
\( \overrightarrow{c}=(13;6) \)
\( \overrightarrow{c_1}=(13;6);\ \overrightarrow{c_2}=(-13;-6) \)
\( \overrightarrow{c}=(13k;6k);\ k\in\mathbb{R} \)
\( \overrightarrow{c}=(-13;-6) \)
1003030603
Level:
B
Let \( \overrightarrow{v}=(12;5) \). Find all the vectors \( \overrightarrow{u} \) that are perpendicular to the vector \( \overrightarrow{v} \) and have the length of \( 26 \).
\( \overrightarrow{u_1} =(10;-24);\ \overrightarrow{u_2}=(-10; 24) \)
\( \overrightarrow{u}=(10;-24) \)
\( \overrightarrow{u_1}=\frac12 (5;-12);\ \overrightarrow{u_2}=\frac12 (-5; 12) \)
\( \overrightarrow{u_1}=26\cdot(5;-12);\ \overrightarrow{u_2}=26\cdot(-5; 12) \)
1103030602
Level:
B
Let \( ABCDV \) be a right pyramid with a square base, such that its opposite edges contain a right angle (see the picture).
Specify the missing coordinate of the apex \( V \).
\( m=2\sqrt2 \)
\( m=-2\sqrt2 \)
\( m=4\sqrt2 \)
\( m=\sqrt2 \)
1103030601
Level:
B
In the cube \( ABCDEFGH \) find the angle \( \varphi \) between the vectors \( \overrightarrow{b}=\overrightarrow{EB} \) and \( \overrightarrow{a}=\overrightarrow{AK} \), where \( K \) is the midpoint of \( HG \). Round \( \varphi \) to the nearest degree.
Help: Choose the appropriate coordinate system.
\( \varphi\doteq 104^{\circ} \)
\( \varphi\doteq 76^{\circ} \)
\( \varphi\doteq 100^{\circ} \)
\( \varphi\doteq 80^{\circ} \)
1103024310
Level:
A
The picture shows the triangle \( KLM \) with indicated vectors \( \overrightarrow{a} \), \( \overrightarrow{b} \), \( \overrightarrow{c} \) in a coordinate system. What are the vector coordinates \( \overrightarrow{b} \)? Express \( \overrightarrow{b} \) as a linear combination of \( \overrightarrow{a} \) and \( \overrightarrow{c} \).
\( \overrightarrow{b} = \left(1;3;4.5\right);\ \overrightarrow{b} = \frac12\overrightarrow{a} + \frac12\overrightarrow{c} \)
\( \overrightarrow{b} = \left(3;1;4.5\right);\ \overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c} \)
\( \overrightarrow{b} = \left(1;3;4.5\right);\ \overrightarrow{b} = \overrightarrow{a} + \overrightarrow{c} \)
\( \overrightarrow{b} = \left(3;1;4.5\right);\ \overrightarrow{b} = \frac12\overrightarrow{a} + \frac12\overrightarrow{c} \)
1103024309
Level:
A
Given the vectors \( \overrightarrow{a} \), \( \overrightarrow{b} \), \( \overrightarrow{c} \) shown in the picture, express a vector \( \overrightarrow{b} \) as a linear combination of vectors \( \overrightarrow{a} \) and \( \overrightarrow{c} \).
\( \overrightarrow{b} = 2\overrightarrow{a} + \overrightarrow{c} \)
\( \overrightarrow{b} = 2\overrightarrow{a} - \overrightarrow{c} \)
\( \overrightarrow{b} = -2\overrightarrow{a} + \overrightarrow{c} \)
\( \overrightarrow{b} = -2\overrightarrow{a} - \overrightarrow{c} \)