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1. Find the angle between the lines whose direction ratios are a,b,c and b–c, c–a, a–b.

2. Find the equation of a line parallel to x-axis and passing through the origin.

3. If the lines \(\frac{x-1}{-3}=\frac{y-2}{2k}=\frac{z-3}{2}\) and \(\frac{x-1}{3k}=\frac{y-1}{1}=\frac{z-6}{-5}\) are perpendicular, find the value of k.

4. Find the shortest distance between lines

\(\vec{r}=6\hat{i}+2\hat{j}+2\hat{k}+\lambda\left(\hat{i}-2\hat{j}+2\hat{k}\right)\) and \(\vec{r}=-4\hat{i}-\hat{k}+\mu\left(3\hat{i}-2\hat{j}-2\hat{k}\right)\).

5. Find the vector equation of the line passing through the point (1,2,–4) and perpendicular to the two lines:

\(\frac{x-8}{3}=\frac{y+19}{-16}=\frac{z-10}{7}\) and \(\frac{x-15}{3}=\frac{y-29}{8}=\frac{z-5}{-5}\).