Some examples of problems involving lines and planes

These problems are based on queries I've received by email, so I think the solutions could be helpful to a lot of people. You might like to have a go at solving them yourself first before looking at my answers. They all depend on the previous vector pages so if you are a bit shaky on vectors you should look at these first.
I think that the clearest way to show vectors in working on screen is as row vectors. So, for example, I shall write 2i + 3j - 4k as (2, 3, -4).

Problem
If u = (-3, 2, 1) and v = (5, 0, -3) and w = (6, 1, -4) is it possible to find a scalar t such that u + tv is parallel to w?
Look at the solution to this problem

Problem
Find out if the line r = (1, 3, 8) + t (-2, 5, 7) is parallel to the plane 3x + 4y - 2z = 1.
Look at the solution to this problem

Problem
Where does the line r = (1, 2, -5) + t (2, -3, 1) meet the plane 2x + 5y - 3z = 6?
Look at the solution to this problem

Problem
Find the vector equation of the line in which the 2 planes 2x - 5y + 3z = 12 and 3x + 4y - 3z = 6 meet.
Look at the solution to this problem

Problem
Finding the shortest distance between a point and a line.
Look at the solution to this problem

Problem
How can we find the shortest distance between two skew lines?
Look at the solution to this problem

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