Points and vectors

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

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

9000101802

Level: 
B
Among vectors \(\vec{u} = \left (- \frac{2} {\sqrt{2}};2\sqrt{2}\right )\), \(\vec{v} = (-5;10)\), \(\vec{w} = (2.5;-5)\), \(\vec{r} = (-3.5;6)\) find the vector which is not parallel to the vector \(\vec{a} = (1;-2)\).
\(\vec{r}\)
\(\vec{w}\)
\(\vec{v}\)
\(\vec{u}\)