List of top Mathematics Questions on Three Dimensional Geometry asked in PSEB XII

Find the shortest distance between the following pairs of lines: \( \vec{r} = \hat{i} - 4\hat{j} + 5\hat{k} + \mu(5\hat{i} + 9\hat{j} + \hat{k}) \) and \( \vec{r} = 2\hat{i} + 8\hat{j} - 6\hat{k} + \lambda(3\hat{i} - 2\hat{j} + \hat{k}) \).
Find the angle between the lines \( \vec{r} = 3\hat{i} + 8\hat{j} + 3\hat{k} + \mu(\hat{i} + 2\hat{j} - \hat{k}) \) and \( \vec{r} = -3\hat{i} + 9\hat{j} - \hat{k} + \lambda(5\hat{i} + 3\hat{j} + 4\hat{k}) \).
Direction ratios of the straight line \( \vec{r} = 2\hat{i} - 3\hat{j} + \hat{k} + m(9\hat{i} - 2\hat{j} + 5\hat{k}) \) are:
Vector equation of the line \( \frac{x-5}{-4} = \frac{y-3}{5} = \frac{z+3}{-8} \) is: