Points and vectors

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

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 \(\left |\vec{w}\right | = 2\).
\(\vec{w} = \left (\frac{2\sqrt{11}} {11} ; \frac{6\sqrt{11}} {11} ; \frac{2\sqrt{11}} {11} \right )\), \(\vec{w} = \left (-\frac{2\sqrt{11}} {11} ;-\frac{6\sqrt{11}} {11} ;-\frac{2\sqrt{11}} {11} \right )\)
\(\vec{w} = (-1;-3;-1)\), \(\vec{w} = (1;3;1)\)
\(\vec{w} = \left (-\frac{1} {2};-\frac{3} {2};-\frac{1} {2}\right )\), \(\vec{w} = \left (\frac{1} {2}; \frac{3} {2}; \frac{1} {2}\right )\)
\(\vec{w} = \left (\frac{2\sqrt{2}} {3} ; \frac{3\sqrt{2}} {2} ; \frac{2\sqrt{2}} {3} \right )\), \(\vec{w} = \left (-\frac{2\sqrt{2}} {3} ;-\frac{3\sqrt{2}} {2} ;-\frac{2\sqrt{2}} {3} \right )\)