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 \(\left(\vec{a}\times\vec{b}\right)\cdot\vec{c}\).
\(\vec{a}\times\vec{b}=(6,-1,3), \left(\vec{a}\times\vec{b}\right)\cdot\vec{c}=-2\)
\(\vec{a}\times\vec{b}=8, \left(\vec{a}\times\vec{b}\right)\cdot\vec{c}=(-8,16,16)\)
\(\vec{a}\times\vec{b}=(-6,1,-3), \left(\vec{a}\times\vec{b}\right)\cdot\vec{c}=2\)
\(\vec{a}\times\vec{b}=\sqrt{46}, \left(\vec{a}\times\vec{b}\right)\cdot\vec{c}=2\)

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.
\(\left [2\pm \frac{\sqrt{3}} {2} , \frac{3} {2} \mp \sqrt{3}\right ]\)
\(\left [1\pm \frac{\sqrt{3}} {2} , \frac{1} {2} \mp \sqrt{3}\right ]\)
\(\left [2\pm \frac{\sqrt{2}} {2} , \frac{3} {2} \mp \sqrt{2}\right ]\)
\(\left [1\pm \frac{\sqrt{2}} {2} , \frac{1} {2} \mp \sqrt{2}\right ]\)

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.
\(68^{\circ }\)
\(75^{\circ }\)
\(44^{\circ }\)
\(61^{\circ }\)