1.

Find the Cartesian equations of the following planes whose vector equations are: `vecr.[(s-2t)hati+93-t)hatj+(2s+t)hatk]=15`

Answer» (a) Given vector equation of plane is,
`vecr.(hati+hatj-hatk) = 2"……"(1)`
Put `hatr = xhati+yhatj+zhatk` in equation (1),
`(xhati+yhatj+zhatk).(hati+hatj-hatk) = 2`
`rArr x+y-z = 2`
Which is the required cartesian equation of the plane.

(b) Given vector equation of the plane is ,
`vecr.(2hati+3hatj-4hatk) = 1 "....."(2)`
put `vecr - x hati+ yhatj+ zhatk` in equation (2),
`(xhati+yhatj+zhatk) (2hati+3hatj-4hatk) = 1`
`2x+3y-4z = 1`
which is the required cartesian equation of the plane.
(c) Given equation of plane is :
`vecr.{(s-2t)hati+(3-t)hatj+(2s+t)hatk} = 15 "......."(3)`
Put `vecr = xhati+yhatj+zhatk.{(s-2t)hati+(3-t)hatj+(2s+t)hatk} = 15`
`rArr x(s-2t)+y(3-t)+z(2s+t)-15=0`
Which is the required cartesian equation of the plane.


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