9000108806
Level:
B
Find the coordinate $y$ which ensures that the vectors \(\vec{u} = (-6;y;3)\) and
\(\vec{v} = (12;4;4)\) are perpendicular.
\(15\)
\(12\)
\(\sqrt{5}\)
\(\frac{5}
{3}\)
Points and vectors
9000101804
Level:
A
In the following list identify a valid relation involving the vectors
\(\vec{a} = (2;-3)\),
\(\vec{b} = (1;3)\) and
\(\vec{c} = (5;-3)\).
\(\vec{c} = 2\vec{a} +\vec{ b}\)
\(\vec{b} = \frac{1}
{2}\vec{a} +\vec{ c}\)
\(2\vec{a} +\vec{ b} +\vec{ c} =\vec{ o}\)
\(\vec{a} = \frac{1}
{2}\vec{b} +\vec{ c}\)
9000101805
Level:
B
Find the vector \(\vec{v}\) such that
the length of this vector is \(5\)
and \(\vec{v}\) is perpendicular
to the vector \(\vec{u} = (-1;0.75)\).
\(\vec{v} = (3;4)\)
\(\vec{v} = (3;-4)\)
\(\vec{v} = (4;-3)\)
\(\vec{v} = (5;0)\)
9000101806
Level:
B
Find the value of the parameter \(a\)
which ensures that the vectors \(\vec{u} = (3;a;-2)\)
and \(\vec{v} = (-6;4;a - 3)\)
are perpendicular.
\(a = 6\)
\(a = 12\)
\(a = -6\)
\(a = 3\)
9000101807
Level:
B
Given points \(A = [1;1]\),
\(B = [5;2]\) and
\(C = [8;7]\), find the
angle \(\measuredangle ABC\).
\(135^{\circ }\)
\(26.5^{\circ }\)
\(30^{\circ }\)
\(60^{\circ }\)
9000101808
Level:
B
Consider a parallelogram \(ABCD\)
with \(A = [1;3]\),
\(B = [2;-1]\) and
\(C = [5;1]\). Let
\(S\) be the center of
the diagonal \(BD\).
Find the vector \(\overrightarrow{AS } \).
\(\overrightarrow{AS } = (2;-1)\)
\(\overrightarrow{AS } = (2;1)\)
\(\overrightarrow{AS } = (1;3)\)
\(\overrightarrow{AS } = (-2;1)\)
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.
\(X_{1} = [2;0],\ X_{2} = [-2;0]\)
\(X = [2;0]\)
\(X = [8;0]\)
\(X_{1} = [2;0],\ X_{2} = [-4;0]\)
9000101802
Level:
B
Among vectors \(\vec{u} = \left (- \frac{2}
{\sqrt{2}};2\sqrt{2}\right )\),
\(\vec{v} = (-5;10)\),
\(\vec{w} = (2.5;-5)\),
\(\vec{r} = (-3.5;6)\)
find the vector which is not parallel to the vector
\(\vec{a} = (1;-2)\).
\(\vec{r}\)
\(\vec{w}\)
\(\vec{v}\)
\(\vec{u}\)
9000100707
Level:
B
Consider points \(A = [-2;-1]\),
\(B = [1;y]\),
\(C = [3;-4]\). Find the coordinate
\(y\) which ensures
that the vectors \(\overrightarrow{AB } \)
and \(\overrightarrow{AC } \)
are perpendicular.
\(y = 4\)
\(y = -4\)
\(y = 0.8\)
\(y = -0.8\)