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The reflection of the point vec a in ...

The reflection of the point ` vec a` in the plane ` vec rdot vec n=q` is a. ` vec a+(( vec q- vec adot vec n))/(| vec n|)` b. ` vec a+2((( vec q- vec adot vec n))/(| vec n|^2)) vec n` c. ` vec a+(2( vec q+ vec adot vec n))/(| vec n|^2) vec n` d. none of these

A

`veca+((vecq-veca.vecn))/(|vecn|)`

B

`veca+2(((vecq-veca.vecn))/(|vecn|^(2)))vecn`

C

`veca+(2(vecq-veca.vecn))/(|vecn|)vecn`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
b
Given plane is `vecr.vecn=q`
Let the image of `A(veca)` in the plane be `B(vecb)`.

Equation of AC is `vecr=veca+lamdavecn`
(`because` AC is normal to the plane) (ii)
Solving (i) and (ii), we get
`(veca+lamdavecn).vecn=q`
or `lamda=(q-veca.vecn)/(|vecn|^(2)).vecn`
But `vec(OC)=(veca+vecb)/(2)`
`becauseveca+((q-veca.vecn)vecn)/(|vecn|^(2))=(veca+vecb)/(2)`
or `vecb=veca+2((q-veca.vecn)/(|vecn|^(2)))`
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