1.

Let L_1 and L_2 denote the lines r=hati+lambda(-hati+2hatj+2hatk).lambdainR. and r=mu(2hati-hatj+2hatk),muinR respectively. If L_3 is a line whichis perpendicular to both L_1 and L_2 and cuts both of them, then which of the following options describes (s)L_3 ?

Answer»

`r=(2)/(9)(2hati-HATJ+2hatk)+t(2hati+2hatj-hatk),t in R`
`r=(1)/(3)(2hati+hatk)+t(2hati+2hatk),t in R`
`r=(2)/(9)(4hati+hatj+hatk)+t(2hati+2hatj-hatk),t in R`
`r=t(2hati+2hatj-hatk),t in R`

Solution :Given lines
`L_i:r=hati+lambda(-hati+2hatj+2hatk),lambda in R and`
`L_2:r=mu(2hati-hatj+2hatj+2hatk),mu in R`
and since line `L_3` is perpendicular to both lines `L_1` and `L_2`.
Then a vector along `L_3` will be,
`|{:(hati,,hatj,,hatk),(-1,,2,,2),(2,,-1,,2):}|=hati(4+2)-hatj(-2-4)+hatk(1-4) =6hati+6hatj-3hatk=3(2hati+2hatj-hatk)`.........(i)
Now, let a general POINT on line `L_1`.
`P(1-lambda ,2lambda,2lambda)` and on line `L_2`.
as `Q(2mu,-mu,2mu) ` and let P and Q
are poitn of intersection of lines `L_1,L_3 and L_2,L_3`, so direction ratio's of `L_3`.
`(2mu+lambda-1,-mu-2lambda,2mu-2lambda)`
Now, `(2mu+lambda-1)/(2)=(-mu-2lambda)/(2)=(2mu-2lambda)/(-1)` [from Eqs. (i) and (ii)]
`rArr lambda (1)/(9) and mu=(2)/(3)`
So, `P((8)/(9),(2)/(9),(2)/(9)) and Q ((4)/(9),(2)/(9),(4)/(9))`
Now, we can take EQUATION of line `L_3` as
`r=a+t(2hati+2hatj-hatk)`, where a is position vector of any point on line`L_3` and possible vector of a are.
`((8)/(9)hati+(2)/(9)hatj+(2)/(9)hatk)or ((4)/(9)hati+(2)/(9)hatj+(4)/(9)hatk)or ((2)/(3)hati+(1)/(3)hatk)`
Hence, OPTIONS (a), (b) and (c) are correct.


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