Points and Vectors

9000100708

Level:

Given points \(A = [-2,-1]\), \(B = [x,-3]\), \(C = [4,-4]\), find the coordinate \(x\) which ensures that the vectors \(\overrightarrow{AB } \) and \(\overrightarrow{AC } \) are parallel.

\(x = 2\)

\(x = 7\)

\(x = -3\)

\(x = 3\)

9000100709

Level:

Find the value of the parameter \(z\) so that the vector \(\vec{w} = (8,2,z)\) is perpendicular to the vectors \(\vec{a} = (1,2,-3)\) and \(\vec{b} = (-1,2,1)\).

\(z = 4\)

\(z = -12\)

\(z = 2\)

\(z = -4\)

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 \(\vec{a} = (1,-2)\).

\(\vec{r}\)

\(\vec{w}\)

\(\vec{v}\)

\(\vec{u}\)

9000101805

Level:

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:

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:

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:

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 \(\overrightarrow{AS } \).

\(\overrightarrow{AS } = (2,-1)\)

\(\overrightarrow{AS } = (2,1)\)

\(\overrightarrow{AS } = (1,3)\)

\(\overrightarrow{AS } = (-2,1)\)

9000108701

Level:

Find all vectors which are perpendicular to the vector \(\vec{u} = (3,4)\) and have the length equal to \(1\).

\(\left (\frac{4} {5},-\frac{3} {5}\right )\), \(\left (-\frac{4} {5}, \frac{3} {5}\right )\)

\(\left (\frac{4} {7},-\frac{3} {7}\right )\), \(\left (-\frac{4} {7}, \frac{3} {7}\right )\)

\(\left ( \frac{1} {\sqrt{10}},- \frac{3} {\sqrt{10}}\right )\), \(\left (- \frac{1} {\sqrt{10}}, \frac{3} {\sqrt{10}}\right )\)

\(\left (\frac{4} {5}, \frac{3} {5}\right )\), \(\left (-\frac{4} {5},-\frac{3} {5}\right )\)

9000108702

Level:

A square has its center in \([1,4]\) and a vertex in \([-1,2]\). Find the remaining vertices of the square.

\([3,6]\), \([-1,6]\), \([3,2]\)

\([3,6]\), \([-1,5]\), \([3,1]\)

\([3,6]\), \([-2,6]\), \([4,2]\)

\([3,6]\), \([-1,5]\), \([3,2]\)

9000108703

Level:

Given points \(A = [1,3]\), \(C = [4,3]\), \(B = [x,2]\), find the value of the parameter \(x\) which ensures that the vector \(AB\) is perpendicular to the vector \(AC\).

\(1\)

\(2\)

\(- 1\)

\(0\)

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