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Distance of the point P( vec p) from the...

Distance of the point `P( vec p)` from the line ` vec r= vec a+lambda vec b` is a. `|( vec a- vec p)=((( vec p- vec a)dot vec b) vec b)/(| vec b|^2)|` b. `|( vec b- vec p)=((( vec p- vec a)dot vec b) vec b)/(| vec b|^2)|` c. `|( vec a- vec p)=((( vec p- vec b)dot vec b) vec b)/(| vec b|^2)|` d. none of these

A

`|(veca=vecp)+(((vecp-veca).vecb)vecb)/(|vecb|^(2))|`

B

`|(vecb-vecp)+(((vecp-veca).vecb)vecb)/(|vecb|^(2))|`

C

`|(veca-vecp)+(((vecp-veca).vecb)vecb)/(|vecb|^(2))|`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
c
Let `P(vecq)` be the foot of altitude drawn from
`P(vecp)` to line `vecr=veca+lamdavecb`
`implies(vecq-vecp).vecb=0andvecq=veca+lamdavecb`
`implies(veca+lamdavecb-vecp).vecb=0`
or `(veca-vecp).vecb+lamda|vecb|^(2)=0`
or `lamda=((vecp-veca).vecb)/(|veca|^(2))`
`impliesvecq-vecp=veca+(((vecp-veca).vecb)vecb)/(|vecb|^(2)`
`implies|vecq-vecp|=|(veca-vecp)+(((vecp-veca).vecb)vecb)/(|vecb|^(2))|`
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