9000100703
Level:
A
Given the vectors \(\vec{a}\),
\(\vec{b}\), and
\(\vec{c}\), find
\(2\vec{a} + 3\vec{b} -\vec{ c}\).
\((13;-5)\)
\((7;-5)\)
\((7;7)\)
\((7;0)\)
Points and vectors
9000100704
Level:
A
Given vectors \(\vec{a} = (2;2;-3)\),
\(\vec{b} = (-1;0;1)\),
\(\vec{c} = (0;-2;1)\), find the length
of the vector \(\vec{u} =\vec{ a} - 2\vec{b} + 3\vec{c}\).
\(|\vec{u}| = 6\)
\(|\vec{u}| = 5\)
\(|\vec{u}| = 4\)
\(|\vec{u}| = 3\)
9000100705
Level:
A
Given vectors \(\vec{a} = (1;y;3)\) and
\(\vec{b} = (2;-1;-2)\), find the value of coordinate
\(y\) so
that the vector \(\vec{u} = (-4;-1;12)\) is a
linear combination of \(\vec{a}\)
and \(\vec{b}\).
\(y = -2\)
\(y = 1\)
\(y = -1\)
\(y = 3\)
9000100710
Level:
A
Given points \(A = [-3;2]\) and
\(B = [1;y]\), find the values of
\(y\) which ensure the
length of the vector \(\overrightarrow{AB } \)
equals \(5\).
\(y_{1} = -1\),
\(y_{2} = 5\)
\(y_{1} = -1\),
\(y_{2} = 1\)
\(y_{1} = 1\),
\(y_{2} = 5\)
\(y_{1} = 5\),
\(y_{2} = -5\)
9000101801
Level:
A
Among vectors \(\vec{a} = (-1;2;0)\),
\(\vec{b} = (2;1;2)\),
\(\vec{c} = (1;3;0)\) and
\(\vec{d} = (-3;0;0)\) find a
pair of vectors with equal length.
\(\vec{b}\),
\(\vec{d}\)
\(\vec{a}\),
\(\vec{c}\)
\(\vec{a}\),
\(\vec{d}\)
\(\vec{b}\),
\(\vec{c}\)
9000101803
Level:
A
In the following list identify a pair of points
\(C\),
\(D\) such that the
vector \(\overrightarrow{CD } \) is not
equal to the vector \(\overrightarrow{AB } \)
where \(A = [1;3;-2]\)
and \(B = [-2;4;3]\).
\(C = [1;-2;3],\ D = [-2;-1;-2]\)
\(C = [6;1;-4],\ D = [3;2;1]\)
\(C = [-3;5;7],\ D = [-6;6;12]\)
\(C = [-3;8;14],\ D = [-6;9;19]\)
9000101804
Level:
A
In the following list identify a valid relation involving the vectors
\(\vec{a} = (2;-3)\),
\(\vec{b} = (1;3)\) and
\(\vec{c} = (5;-3)\).
\(\vec{c} = 2\vec{a} +\vec{ b}\)
\(\vec{b} = \frac{1}
{2}\vec{a} +\vec{ c}\)
\(2\vec{a} +\vec{ b} +\vec{ c} =\vec{ o}\)
\(\vec{a} = \frac{1}
{2}\vec{b} +\vec{ c}\)
9000101809
Level:
A
Given point \(A = [3;2]\),
find all the points \(X\)
on the \(y\)-axis
such that \(|AX| = 5\).
\(X_{1} = [0;-2],\ X_{2} = [0;6]\)
\(X_{1} = [0;-6],\ X_{2} = [0;2]\)
\(X_{1} = [0;-6],\ X_{2} = [0;-2]\)
\(X_{1} = [0;2],\ X_{2} = [0;6]\)
9000101810
Level:
A
Given points \(A = [1;2]\)
and \(B = [4;4]\), find the
point \(X\) on the
\(x\)-axis such that
the distance from \(X\)
to \(B\) is a double of
the distance from \(X\)
to \(A\).
Find all solutions of the problem.
\(X_{1} = [2;0],\ X_{2} = [-2;0]\)
\(X = [2;0]\)
\(X = [8;0]\)
\(X_{1} = [2;0],\ X_{2} = [-4;0]\)
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 \( 26 \).
\( \overrightarrow{u_1} =(10;-24);\ \overrightarrow{u_2}=(-10; 24) \)
\( \overrightarrow{u}=(10;-24) \)
\( \overrightarrow{u_1}=\frac12 (5;-12);\ \overrightarrow{u_2}=\frac12 (-5; 12) \)
\( \overrightarrow{u_1}=26\cdot(5;-12);\ \overrightarrow{u_2}=26\cdot(-5; 12) \)