# A line with direction ratios < 2, 2, 1 > intersects the lines at the points P and Q respectively. Find the length and the equation of the intercept PQ.

Let a be the common value of
we can rewrite there equation under the following equivalent parametric form

In the same way, the generic point on the second straight line is

Then you just have to express the proportionality of with the given, vector, yielding the following system of 2 equations and 2 unknowns.

giving
It remains to  placing there values in (1) and (2) to get the coordinates of

The length of PQ

The equation of intercept PQ is

is the required equation

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