Question

If the points A, B, C and D have position vectors $\overrightarrow{a},2\,\overrightarrow{a}+\overrightarrow{b},4\,\overrightarrow{a}+2\,\overrightarrow{b}$ and $5\,\overrightarrow{a}+4\,\overrightarrow{b},$ respectively. Then the three collinear points are

• A. A, B and C
• B. A, C and D
• C. A, B, and D
• D. B, C and D.

Answer 1

Answer: (C)

$\overrightarrow{AB}=2\,\vec{a}+\vec{b}-\vec{a}=\vec{a}+\vec{b}$
$\overrightarrow{AC}=4\,\vec{a}+2\,\vec{b}-\vec{a}=3\,\vec{a}+2\,\vec{b}$
$\overrightarrow{AD}=5\,\vec{a}+4\,\vec{b}-\vec{a}=4\left(\vec{a}+\vec{b}\right)$
Since $\overrightarrow{AD}=4\,\overrightarrow{AB}$
$\therefore A$, $B$, $D$ are collinear.