Question

If $2\hat{i}+3\hat{j}+4\hat{k}$ and $\hat{i}-4\hat{j}+7\hat{k}$ are the sides of a parallelogram, then is the vector representing its diagonal.

• A. $3\hat{i}-\hat{j}+11\hat{k}$
• B. $3\hat{i}+7\hat{j}+7\hat{k}$
• C. $\hat{i}-7\hat{j}+11\hat{k}$

Answer 1

The correct option is A $3\hat{i}-\hat{j}+11\hat{k}$

Let $\overrightarrow{a}=2\hat{i}+3\hat{j}+4\hat{k}$ and $\overrightarrow{b}=\hat{i}-4\hat{j}+7\hat{k}$

If $\overrightarrow{a}$ and $\overrightarrow{b}$ are the sides of a parallelogram, then one of the diagonals is $\overrightarrow{a}+\overrightarrow{b}$ and the other diagonal is $\overrightarrow{b}-\overrightarrow{a}$

$\overrightarrow{a}+\overrightarrow{b}=(2\hat{i}+3\hat{j}+4\hat{k})+(\hat{i}-4\hat{j}+7\hat{k})$
$=(2+1)\hat{i}+(3-4)\hat{j}+(4+7)\hat{k}$
$=3\hat{i}-\hat{j}+11\hat{k}$

$\overrightarrow{b}-\overrightarrow{a}=(\hat{i}-4\hat{j}+7\hat{k})-(2\hat{i}+3\hat{j}+4\hat{k})$
$=(1-2)\hat{i}+(-3-4)\hat{j}+(7-4)\hat{k}$
$=-\hat{i}-7\hat{j}+3\hat{k}$

Thus, among the options, the diagonal vector is $3\hat{i}-\hat{j}+11\hat{k}$