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# P3 Vectors - quite quick watch

1. Hiya, could someone do the following question for me?:

Find the shortest distance of the point (6,6,-1) from the line with equation r = 2i = j - 3k + t(i+2j-k)

I know shortest distance is the perpendicular distance and so I tried to calculate that. I got around 3sumthing for my answer (i've lost sheet of paper with my working and answer), but the book says sqrt21? Could someone show me? Thanks

And good luck for 2moz every!
2. Ok, lets call the nearest point on the line P, and this can be written in the form
P = (2, 1, -3) + t[1,2,-1]

The vector from (6,6,-1) to P is perpendicular to the line, so dot product is 0
And the vector is (6,6,-1)-P = (6-2-t, 6-1-2t, -1+3+t)

So
[1,2,-1] . [6-2-t, 6-1-2t, -1+3+t] = 0
0 = 6-2-t + 12-2-4t + 1-3-t
6t = 12
t = 2

Sub this back to find Q = (4, 5, -5)

Distance is therefore sqrt(2^2 + 1 + 4^2) = sqrt(21).

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Updated: June 8, 2004
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