Points and Vectors

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 \(\vec{c} = (5;-3)\).
\(\vec{c} = 2\vec{a} +\vec{ b}\)
\(\vec{b} = \frac{1} {2}\vec{a} +\vec{ c}\)
\(2\vec{a} +\vec{ b} +\vec{ c} =\vec{ o}\)
\(\vec{a} = \frac{1} {2}\vec{b} +\vec{ c}\)

9000101808

Level: 
B
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 \(\overrightarrow{AS } \).
\(\overrightarrow{AS } = (2;-1)\)
\(\overrightarrow{AS } = (2;1)\)
\(\overrightarrow{AS } = (1;3)\)
\(\overrightarrow{AS } = (-2;1)\)