Question

If $A$ and $B$ be points with position vectors $a$ and $b$ with respect to origin $O$.

If the point $C$ on $OA$ is such that $2AC=CO$, $CD$ is parallel to $\left|CD\right|=3\left|OB\right|$, then $\left|AD\right|$ is equal to?

• A. $2b-\frac{1}{2}a$
• B. $3b-\frac{1}{3}a$
• C. $b-\frac{1}{2}a$
• D. None of these

Answer 1

The correct option is B

$3b-\frac{1}{3}a$

Explanation for the correct option:

Find the value of $\left|AD\right|$:

Given, $2AC=CO$

$\Rightarrow$$2(c–a)=-c$

$\Rightarrow$ $3c=2a$

$\Rightarrow$ $c=\frac{2a}{3}$

Also given, $CD\parallel OB$

$\Rightarrow$ $a–c=kb$

$\Rightarrow$$|d–c|=3\left|b\right|$

$\Rightarrow$ $k\left|b\right|=3\left|b\right|$

$\Rightarrow$ $k=3$

and $d=c+3b$

$\begin{array}{rcl}\therefore AD& =& d–a\\ & =& 3b+c–a\\ & =& \mathbf{3}\mathit{b}\mathbf{}\mathbf{–}\mathbf{}\frac{\mathbf{a}}{\mathbf{3}}\end{array}$

Hence, Option ‘B’ is Correct.