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Question
Find the length of the perpendicular drawn from the point P(3, 2, 1) to the line `overliner = (7hati + 7hatj + 6hatk) + λ(-2hati + 2hatj + 3hatk)`
Solution
The length of the perpendicular is same as the distance of P from the given line.
The distance of point `P(overlineα)` from the line `overliner = overlinea + λoverlineb` is `sqrt(|overlineα - overlinea|^2 - [[(overlineα - overlinea).overlineb)/|overlineb|]^2`
Here `overlineα = 3hati + 2hatj + hatk, overlinea = 7hati + 7hatj + 6hatk, overlineb = -2hati + 2hatj + 3hatk`
∴ `overlineα - overlinea = (3hati + 2hatj + hatk) -(7hati + 7hatj + 6hatk) = -4hati - 5hatj - 5hatk`
`|overlineα - overlinea| = sqrt((-4)^2 + (-5)^2 + (-5)^2)`
= `sqrt(16 + 25 + 25)`
= `sqrt(66)`
`(overlineα - overlinea).overlineb = (-4hati - 5hatj - 5hatk).(-2hati + 2hatj + 3hatk)`
= 8 – 10 – 15
= –17
`|overlineb| = sqrt((-2)^2 + (2)^2 + (3)^2)`
= `sqrt(17)`
The require length = `sqrt(|overlineα - overlinea|^2 - [[(overlineα - overlinea).overlineb)/|overlineb|]^2`
= `sqrt(66 - |(-17)/sqrt(17)|^2`
= `sqrt(66 - 17)`
= `sqrt(49)`
= 7 unit
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