Collinear points and parallel lines with vectors Problem set A

In triangle ABC, \(\overrightarrow{AB}=2\mathbf{a}\) and \(\overrightarrow{AC}=5\mathbf{b}\). The point D lies on BC with BD:DC = 3:2. The point E is such that \(\overrightarrow{BE}=\frac{3}{2}\overrightarrow{AC}\). Show that A, D and E lie on a straight line.

In triangle OAB, \(\overrightarrow{OA}=4\mathbf{a}\) and \(\overrightarrow{OB}=6\mathbf{b}\). M is the midpoint of OB and N is the midpoint of AB. Show that MN is parallel to OA.

Show that the points A(1, 2, −1), B(4, −1, 5) and C(7, −4, 11) are collinear.

A triangular sail ABC has a frame, \(\overrightarrow{AB}=3\mathbf{a}\) and \(\overrightarrow{AC}=4\mathbf{b}\). The point D lies on BC with BD:DC = 1:3. The point E is such that \(\overrightarrow{BE}=\frac{1}{3}\overrightarrow{AC}\). Show that A, D and E lie on a straight line.

A garden bed ABCD is a trapezium (lengths in metres), \(\overrightarrow{AB}=10\mathbf{a}\), \(\overrightarrow{DC}=5\mathbf{a}\) and \(\overrightarrow{AD}=2\mathbf{a}+4\mathbf{b}\). M is the midpoint of AB. Show that DM is parallel to CB.

A drone is seen at three positions P(2, −1, 4), Q(5, 2, −2) and R(11, 8, −14) (coordinates in metres). Show that the drone moved in a straight line.

In triangle ABC, \(\overrightarrow{AB}=5\mathbf{a}\) and \(\overrightarrow{AC}=2\mathbf{b}\). The point D lies on BC with BD:DC = 2:3. The point E is such that \(\overrightarrow{BE}=k\overrightarrow{AC}\). Find the value of \(k\) for which A, D and E lie on a straight line.

WXYZ is a quadrilateral, \(\overrightarrow{WX}=8\mathbf{a}\), \(\overrightarrow{ZY}=4\mathbf{a}\) and \(\overrightarrow{YX}=3\mathbf{b}\). The point P is such that WYP is a straight line and Y is the midpoint of WP. Show that WZ is parallel to XP.