Question

Which of the following is the direction vector for the shortest distance between the lines L1 and L2, whose vector equations are$\overrightarrow{V_1}=2\hat{i}+3\hat{j}+\lambda (5\hat{i}+3\hat{j}-3\hat{k})$ and $\overrightarrow{V_2}=\hat{i}+4\hat{j}+\lambda (3\hat{i}+2\hat{j}+\hat{k}).$

• A. $-\hat{i}+\hat{j}$
• B. $-2\hat{i}-\hat{j}+4\hat{k}$
• C. $9\hat{i}-14\hat{j}+\hat{k}$
• D. $-9\hat{i}-14\hat{j}$

Answer 1

The correct option is C $9\hat{i}-14\hat{j}+\hat{k}$

Shortest distance between two vectors lies in the direction of $\overrightarrow{b_1}\times \overrightarrow{b_2}$
Here, $\overrightarrow{b_1}=5\hat{i}+3\hat{j}-3\hat{k}$ and
$\overrightarrow{b_2}=3\hat{i}+2\hat{j}+\hat{k}$
Taking their cross product
$\overrightarrow{b_1}\times \overrightarrow{b_2}=(5\hat{i}+3\hat{j}-3\hat{k})\times (3\hat{i}+2\hat{j}+\hat{k})=9\hat{i}-14\hat{j}+\hat{k}$