Points and Vectors

1103030704

Level: 
A
We are given points \( A = [2,1] \), \( B = [4,-1] \), and \( T = [6,2] \), where point \( T \) is the centroid of triangle \( ABC \). Find the length of the median of triangle \( ABC \) to side \( AC \).
\( |t_b|=\frac{\sqrt{117}}2 \)
\( |t_b|=\frac{\sqrt{45}}2 \)
\( |t_b|=\frac{\sqrt{153}}2 \)
\( |t_b|=\sqrt{117} \)

1103030701

Level: 
A
We are given points \( A = [1,-1,2] \), \( B = [0,5,-3] \), \( S = [2,0,5] \). Point \( S \) is the centre of a parallelogram \( ABCD \). Find the coordinates of vertices \( C \) and \( D \).
\( C = [3,1,8], D = [4,-5,13] \)
\( C = [4,-5,13], D = [3,1,8] \)
\( C = [1,1,3], D = [2,-5,8] \)
\( C = [-3,-1,-8], D = [-4,5,-13] \)

1003020901

Level: 
C
Let there be vectors: \(\vec{a}=(1,3,-1)\), \(\vec{b}=(0,3,1)\), \(\vec{c}=(-1,2,2)\). Find \(\vec{a}\times\vec{b}\) and \(\left(\vec{a}\times\vec{b}\right)\cdot\vec{c}\).
\(\vec{a}\times\vec{b}=(6,-1,3), \left(\vec{a}\times\vec{b}\right)\cdot\vec{c}=-2\)
\(\vec{a}\times\vec{b}=8, \left(\vec{a}\times\vec{b}\right)\cdot\vec{c}=(-8,16,16)\)
\(\vec{a}\times\vec{b}=(-6,1,-3), \left(\vec{a}\times\vec{b}\right)\cdot\vec{c}=2\)
\(\vec{a}\times\vec{b}=\sqrt{46}, \left(\vec{a}\times\vec{b}\right)\cdot\vec{c}=2\)

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.
\(\vec{w} = (0,4)\), \(\vec{w} = (2\sqrt{3},-2)\)
\(\vec{w} = (0,-4)\), \(\vec{w} = (\sqrt{7},-3)\)
\(\vec{w} = (0,4)\), \(\vec{w} = (\sqrt{7},3)\)
\(\vec{w} = (\sqrt{5},\sqrt{11})\), \(\vec{w} = (2\sqrt{3},-2)\)

9000108804

Level: 
B
The point \(A = [3,2]\) is rotated about the center \(B = [1,1]\) by \(60^{\circ }\). Find the coordinate of its final position. Consider both clockwise and counterclockwise direction.
\(\left [2\pm \frac{\sqrt{3}} {2} , \frac{3} {2} \mp \sqrt{3}\right ]\)
\(\left [1\pm \frac{\sqrt{3}} {2} , \frac{1} {2} \mp \sqrt{3}\right ]\)
\(\left [2\pm \frac{\sqrt{2}} {2} , \frac{3} {2} \mp \sqrt{2}\right ]\)
\(\left [1\pm \frac{\sqrt{2}} {2} , \frac{1} {2} \mp \sqrt{2}\right ]\)