Question

The shortest distance between the lines $x = y = z$ and $x=1-y=\frac{z}{0}$ is

• A. $\frac{1}{2}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\frac{1}{\sqrt{3}}$
• D. $\frac{1}{\sqrt{6}}$

Answer 1

Answer: (D)

The lines are $\frac{x}{1}=\frac{y}{1}=\frac{z}{1}$ and $\frac{x}{1}=\frac{y-1}{-1}=\frac{z}{0}$,
Now, $\vec{a}_{1}=0$,
$\vec{b}_{1}=\hat{i}+\hat{j}+\hat{k}$,
$\vec{a}_{2}=\hat{j}$,
$\vec{b}_{2}=\hat{i}-\hat{j}$
$\vec{b}_{1}\times\vec{b}_{2}=\left(\hat{i}+\hat{j}+\hat{k}\right)\times\left(\hat{i}-\hat{j}\right)=\hat{i}+\hat{j}-2\hat{k}$
$S.D. =\frac{\left|\hat{j} \cdot\left(\hat{i}+\hat{j}-2\hat{k}\right)\right|}{\left|\hat{i}+\hat{j}-2\hat{k}\right|}$
$=\frac{1}{\sqrt{6}}$.