The points A A\,A and B B\,B have position vectors (3 −2 1)\begin{pmatrix} 3 \ -2 \ 1 \end{pmatrix}(3 −2 1) and (1 2 −1)\begin{pmatrix} 1 \ 2 \ -1 \end{pmatrix}(1 2 −1) respectively.
Find the exact length of ABABAB.
Find the position vector of the midpoint of ABABAB.
The points P P\,P and Q Q\,Q have position vectors (2 1 3)\begin{pmatrix} 2 \ 1 \ 3 \end{pmatrix}(2 1 3) and (4 −3 5)\begin{pmatrix} 4 \ -3 \ 5 \end{pmatrix}(4 −3 5) respectively. Show that ABPQ ABPQ\,ABPQ is a parallelogram.
