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The distance between the line vec r=(...

The distance between the line ` vec r=(2 hat i-2 hat j+3 hat k)+lambda( hat i- hat j+4 hat k)` and plane ` vec rdot(( hat i+5 hat j+ hat k))=5.`

A

`(10)/(3sqrt3)`

B

`(10)/(9)`

C

`(10)/(3)`

D

`(3)/(10)`

Text Solution

Verified by Experts

The correct Answer is:
a
It is obvious that the given line and point are parallel. Given point on the line is A(2,-2,3). B(0,0,5) is a point on the plane. Therefore,
`vec(AB)=(2-0)hati+(-2-0)hatj+(3-5)hatk` ltbgt Then distance of B from the plane = projection of
`vec(AB)" on vector "hati+5hatj+hatk`
`p=|((2hati-2hatj-2hatk).(hati+5hatj+hatk))/(sqrt(1+25+1))|`
`=|(2-10-2)/(sqrt27)|=(10)/(3sqrt3)`
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