9000108707
Level:
C
Find the area of the parallelogram \(ABCD\)
where \(A = [1;3;2]\),
\(B = [3;4;6]\),
\(D = [2;5;8]\).
\(\sqrt{77}\)
\(8\)
\(2\sqrt{13}\)
\(\sqrt{94}\)
Points and vectors
9000108801
Level:
B
Find the angle between the vector \(\vec{u} = (1;\sqrt{2})\)
and \(\vec{v} = (3;-1)\).
Round to the nearest degree.
\(73^{\circ }\)
\(42^{\circ }\)
\(57^{\circ }\)
\(64^{\circ }\)
9000108802
Level:
B
Given the points \(A = [1;2]\),
\(B = [2;6]\) and
\(C = [3;-1]\), find the interior
angles of the triangle \(ABC\).
Round to the nearest degree.
\(22^{\circ }\),
\(26^{\circ }\),
\(132^{\circ }\)
\(26^{\circ }\),
\(45^{\circ }\),
\(109^{\circ }\)
\(22^{\circ }\),
\(48^{\circ }\),
\(110^{\circ }\)
\(17^{\circ }\),
\(31^{\circ }\),
\(132^{\circ }\)
9000108803
Level:
B
Consider the vector \(\vec{u} = (\sqrt{3};1)\).
Find the vector \(\vec{w}\)
such that \(\left |\vec{w}\right | = 4\) and the
angle between \(\vec{u}\)
and \(\vec{w}\) is
\(60^{\circ }\). Find
all solutions.
\(\vec{w} = (0;4)\),
\(\vec{w} = (2\sqrt{3};-2)\)
\(\vec{w} = (0;-4)\),
\(\vec{w} = (\sqrt{7};-3)\)
\(\vec{w} = (0;4)\),
\(\vec{w} = (\sqrt{7};3)\)
\(\vec{w} = (\sqrt{5};\sqrt{11})\),
\(\vec{w} = (2\sqrt{3};-2)\)
9000108805
Level:
B
Find the angle between \(\vec{u} = (1;-2;3)\)
and \(\vec{v} = (-1;0;2)\).
Round to the nearest degree.
\(53^{\circ }\)
\(27^{\circ }\)
\(60^{\circ }\)
\(46^{\circ }\)
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.
\(15\)
\(12\)
\(\sqrt{5}\)
\(\frac{5}
{3}\)
9000108807
Level:
B
Find the angle between the median \(t_{c}\)
and side \(c\) in
the triangle \(ABC\)
for \(A = [1;2]\),
\(B = [7;-2]\) and
\(C = [6;1]\).
Round to the nearest degree. Hint: In geometry, the median \(t_{c}\)
of the triangle \(ABC\) is the line
segment joining the vertex \(C\)
to the midpoint of the opposing side.
\(60^{\circ }\)
\(50^{\circ }\)
\(43^{\circ }\)
\(71^{\circ }\)
9000108808
Level:
B
Find the angle between the altitude \(v_{c}\)
and side \(b\) in
the triangle \(ABC\)
for \(A = [1;2]\),
\(B = [7;-2]\) and
\(C = [6;1]\).
Round to the nearest degree. Hint: In geometry, the altitude \(v_{c}\)
of the triangle \(ABC\) is the line
segment through the vertex \(C\)
and perpendicular to the line containing the opposite side of the triangle.
\(68^{\circ }\)
\(75^{\circ }\)
\(44^{\circ }\)
\(61^{\circ }\)
9000108706
Level:
B
Find all vectors which are parallel to the vector
\(\vec{u} = (3;-1)\) and have the
length equal to \(1\).
\(\left (\frac{3\sqrt{10}}
{10} ;-\frac{\sqrt{10}}
{10} \right )\),
\(\left (-\frac{3\sqrt{10}}
{10} ; \frac{\sqrt{10}}
{10} \right )\)
\((0;-1)\),
\((0;1)\)
\((-3;1)\),
\((3;-1)\)
\(\left (\frac{3}
{4};-\frac{1}
{4}\right )\),
\(\left (-\frac{3}
{4}; \frac{1}
{4}\right )\)
9000108708
Level:
C
Find the volume of the parallelepiped \(ABCDEFGH\)
where \(A = [1;0;0]\),
\(B = [2;0;0]\),
\(D = [3;-2;0]\),
\(E = [2;1;5]\).
\(10\)
\(12\)
\(15\)
\(20\)