The value of `lamda` for which the vectors `3hati-6hatj+hatkand2hati-4hatj+lamdahatk` parallel, is

Options

A. `(2)/(3)`
B. `(3)/(2)`
C. `(5)/(2)`
D. `(2)/(5)`

Answer

A

Explanation

Since, two vectors are parallel i.e., angle between them is zero.
`:.(3hati-6hatj+hatk).(2hati-4hatj+lamdahatk)=|3hati-6hatj+hatk|.|2hati-4hatj+lamdahatk|`
`[:'veca.vecb=|a|cos0^(@)impliesveca.vecb=|veca|vecb|]`
`implies6+24+lamda=sqrt(9+36+1)sqrt(4+16+lamda^(2))`
`implies30+lamda=sqrt(46)sqrt(20+lamda^(2))`
`implies900+lamda^(2)+60lamda=46(20+lamda^(2))` [on squaring both sides]
`implieslamda^(2)+60lamda-20=920-900`
`implies-45lamda^(2)+60lamda-20=0`
`implies-45lamda^(2)+30lamda+30lamda-20=0`
`implies-15lamda(3lamda-2)+10(3lamda-2)=0`
`implies(10-15lamda)(3lamda-2)=0`
`:.lamda=(2)/(3).(2)/(3)`
Alternate Method
Let `veca=3hati-6hatj+hatkandvecb=2hati-4hatj+lamdahatk`
Since, `veca||vecb`
`implies(3)/(2)=(-6)/(-4)=(1)/(lamda)implieslamda=(2)/(3)`