Points and vectors

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 \(1\).
\(\left (\frac{3\sqrt{10}} {10} ;-\frac{\sqrt{10}} {10} \right )\), \(\left (-\frac{3\sqrt{10}} {10} ; \frac{\sqrt{10}} {10} \right )\)
\((0;-1)\), \((0;1)\)
\((-3;1)\), \((3;-1)\)
\(\left (\frac{3} {4};-\frac{1} {4}\right )\), \(\left (-\frac{3} {4}; \frac{1} {4}\right )\)

9000108704

Level: 
B
Consider a pair of vectors \(\vec{u} = (1;0;-1)\) and \(\vec{v} = (2;-1;1)\). Find all the vectors \(\vec{w}\) which are perpendicular to both \(\vec{u}\) and \(\vec{v}\) and satisfy \(\left |\vec{w}\right | = 2\).
\(\vec{w} = \left (\frac{2\sqrt{11}} {11} ; \frac{6\sqrt{11}} {11} ; \frac{2\sqrt{11}} {11} \right )\), \(\vec{w} = \left (-\frac{2\sqrt{11}} {11} ;-\frac{6\sqrt{11}} {11} ;-\frac{2\sqrt{11}} {11} \right )\)
\(\vec{w} = (-1;-3;-1)\), \(\vec{w} = (1;3;1)\)
\(\vec{w} = \left (-\frac{1} {2};-\frac{3} {2};-\frac{1} {2}\right )\), \(\vec{w} = \left (\frac{1} {2}; \frac{3} {2}; \frac{1} {2}\right )\)
\(\vec{w} = \left (\frac{2\sqrt{2}} {3} ; \frac{3\sqrt{2}} {2} ; \frac{2\sqrt{2}} {3} \right )\), \(\vec{w} = \left (-\frac{2\sqrt{2}} {3} ;-\frac{3\sqrt{2}} {2} ;-\frac{2\sqrt{2}} {3} \right )\)

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.
\(X_{1} = [2;0],\ X_{2} = [-2;0]\)
\(X = [2;0]\)
\(X = [8;0]\)
\(X_{1} = [2;0],\ X_{2} = [-4;0]\)

9000101804

Level: 
A
In the following list identify a valid relation involving the vectors \(\vec{a} = (2;-3)\), \(\vec{b} = (1;3)\) and \(\vec{c} = (5;-3)\).
\(\vec{c} = 2\vec{a} +\vec{ b}\)
\(\vec{b} = \frac{1} {2}\vec{a} +\vec{ c}\)
\(2\vec{a} +\vec{ b} +\vec{ c} =\vec{ o}\)
\(\vec{a} = \frac{1} {2}\vec{b} +\vec{ c}\)