The shortest distance between the lines $\vec{r} = (4\hat{i} - \hat{j}) + \lambda(\hat{i} + 2\hat{j} - 3\hat{k})$ and $\vec{r} = (\hat{i} - \hat{j} + 2\hat{k}) + \mu(\hat{i} + 4\hat{j} - 5\hat{k})$ is...

Let $\vec{r} \cdot (3\hat{i} - 2\hat{j} + 7\hat{k}) = 32$ is the equation of a plane and the line having direction ratios $(5, b, 3)$ is parallel to the plane, then the value of b is...

A line passing through the points $(1, -1, 2)$ and $(2, 0, 1)$ meets the XY plane and the YZ plane at points A and B respectively. The distance AB is equal to...

An airplane can carry a maximum of $250$ passengers. A profit of Rs $1500$ is made on each executive class ticket and a profit of Rs $900$ is made on each economy class ticket. The airline reserves at least $30$ seats for executive class. However at least $4$ times as many passengers prefer to travel by economy class than by executive class. Let $x_1$ be the number of passengers of executive class and $x_2$ be the number of passengers of economy class. Formulate the LPP in order to maximize the profit for the airline...