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Question

Find the shortest distance between the point (2,4,5) and the line x23=y+21=z+12.


Solution


Here, P(2,4,5) is the given point.
x23=y+21=z+12=λ
Any point Q on the line is (3λ+2,λ2,2λ1).

PQ=(3λ+4)^i+(λ6)^j+(2λ+4)^k

For the shortest distance, PQ(3^i+^j+2^k)
(3λ+4)×3+(λ6)×1+(2λ+4)×2=0
14λ+14=0
λ=1

PQ=^i7^j+2^k 
PQ=12+(7)2+22
             =54=36 

Shortest distance between the point (2,4,5) and the line x23=y+21=z+12 is 36 units. 

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