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
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
\(\vec{a} = (1,-2)\).
\(\vec{r}\)
\(\vec{w}\)
\(\vec{v}\)
\(\vec{u}\)
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 \(B = [-2,4,3]\).
\(C = [1,-2,3],\ D = [-2,-1,-2]\)
\(C = [6,1,-4],\ D = [3,2,1]\)
\(C = [-3,5,7],\ D = [-6,6,12]\)
\(C = [-3,8,14],\ D = [-6,9,19]\)