Question

If $\overline{OA}=i+j+k,\overline{AB}=3i-2j+k,\overline{BC}=i+2j-2k$ and $\overline{CD}=2i+j+3k$ then find the vector $\overline{OD}$.

• A. $7i-2j-6k$
• B. $7i+2j+3k$
• C. $7i+2j+5k$
• D. None of these

Answer 1

The correct option is B $7i+2j+3k$

$\overrightarrow{AB}=3\hat{i}-2\hat{j}+\hat{k}$

$\Rightarrow \overrightarrow{OB}-\overrightarrow{OA}=3i-2\hat{j}+\hat{k}$

$\Rightarrow \overrightarrow{OB}=3i-2\hat{j}+\hat{k}+\overrightarrow{OA}$

$\Rightarrow \overrightarrow{OB}=3i-2\hat{j}+\hat{k}+\left(\hat{i}+\hat{j}+\hat{k}\right)$

$\Rightarrow \overrightarrow{OB}=4\hat{i}-\hat{j}+2\hat{k}$

Now,

$\overrightarrow{BC}=\overrightarrow{OC}-\overrightarrow{OB}$

$\Rightarrow \overrightarrow{OC}=\overrightarrow{BC}+\overrightarrow{OB}$

$\Rightarrow \overrightarrow{OC}=\left(i+2j-2k\right)+\left(4\hat{i}-\hat{j}+2\hat{k}\right)$

$\Rightarrow \overrightarrow{OC}=5\hat{i}+\hat{j}$

similarly, $\overrightarrow{CD}=\overrightarrow{OD}-\overrightarrow{OC}$

$\Rightarrow \overrightarrow{OD}=\overrightarrow{CD}+\overrightarrow{OC}$

$\Rightarrow \overrightarrow{OD}=\left(2\hat{i}+\hat{j}+3\hat{k}\right)+\left(5\hat{i}+\hat{j}\right)$

$\Rightarrow \overrightarrow{OD}=7\hat{i}+2\hat{j}+3\hat{k}$