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The position vectors of the points where...

The position vectors of the points where the line `r=hat(i)-hat(j)+hat(k)+t(hat(i)+hat(j)-hat(k))` meets the plane `r*(hat(i)+hat(j)+hat(k))=5`, is

A

`5hat(i)+hat(j)+hat(k)`

B

`5hat(i)+3hat(j)-3hat(k)`

C

`2hat(i)+hat(j)+2hat(k)`

D

`5hat(i)+hat(j)+hat(k)`

Text Solution

Verified by Experts

The correct Answer is:
B
Given equation of line is
`r=hat(i)-hat(j)+hat(k)+(hat(i)+hat(j)-hat(k))`
Plane, `r*(hat(i)+hat(j)+k)=5`
Equation of line and in cartesian from
`(x-1)/(2)=(y+1)/(1)=(z-1)/(-1)` . . . (i)
`and" "x+y+z=5` . . . (ii)
`"Let "(x-1)/(1)=(y+1)/(1)=(z-1)/(1)=lamda`
`rArr" "x=lamda+1,y=lamda-1,a=-lamda+1`
but putting value of x,y and z in Eq. (ii)
`lamda+1+lamda-1-lamda+1=5`
`rArr" "2lamda-lamda+2-1=5`
`rArr" "lamda=4`
so points are `x=5,y=3,z=-3`
Hence, the position vector of the point is `5hat(i)+3hat(j)-3hat(k)`
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