The points A A\,A and B B\,B have position vectors (1−24)\begin{pmatrix} 1 \\ -2 \\ 4 \end{pmatrix}1−24 and (3−52)\begin{pmatrix} 3 \\ -5 \\ 2 \end{pmatrix}3−52 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 (421)\begin{pmatrix} 4 \\ 2 \\ 1 \end{pmatrix}421 and (253)\begin{pmatrix} 2 \\ 5 \\ 3 \end{pmatrix}253 respectively. Show that ABPQ ABPQ\,ABPQ is a parallelogram.
