(i)[5]
Calculate the magnitudes $|\overrightarrow{AB}|$, $|\overrightarrow{BC}|$ and $|\overrightarrow{CD}|$.
(ii)[4]
Find the two possible position vectors of point $D$, given that $D$ lies on the line through $B$ and $C$.
Mathematics 9709 · AS & A Level · Coordinate geometry
Coordinate geometry — practice question
With respect to an origin $O$, the position vectors of points $A$, $B$ and $C$ are given by $\overrightarrow{OA} = \begin{pmatrix} 8 \\ -6 \\ 5 \end{pmatrix}$, $\overrightarrow{OB} = \begin{pmatrix} -10 \\ 3 \\ -13 \end{pmatrix}$ and $\overrightarrow{OC} = \begin{pmatrix} 2 \\ -3 \\ 7 \end{pmatrix}$. A fourth point, $D$, is such that the magnitudes $|\overrightarrow{AB}|$, $|\overrightarrow{BC}|$ and $|\overrightarrow{CD}|$ are, respectively, the first, second and third terms of a geometric progression.