\(The line l has equation r = i + 2j + 3k + μ(2i - j - 2k).\)
The point P has position vector 4i + 2j - 3k. Find the length of the perpendicular from P to l.
Solution
1. Find \(\overrightarrow{PQ}\) for a general point Q on l, e.g., \(-3\mathbf{i} + 6\mathbf{k} + \mu(2\mathbf{i} - \mathbf{j} - 2\mathbf{k})\).
2. Calculate the scalar product of \(\overrightarrow{PQ}\) and a direction vector for l and equate the result to zero.
3. Solve for \(\mu\) and obtain \(\mu = 2\).
4. Carry out a complete method for finding the length of \(\overrightarrow{PQ}\).
5. Obtain answer 3.
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