Statement 1: Distance of point D( 1,0,-1) from the plane of points A( 1,-2,0) , B ( 3, 1,2) and C( -1,1,-1) is `8/sqrt229`
Statement 2: volume of tetrahedron formed by the points A,B, C and D is `sqrt229/ 2`

Options

A. Both the statements are true and statement 2 is the correct explanation for statement 1.
B. Both statements are true but statement 2 is not the correct explanation for statement 1.
C. Statement 1 is true and Statement 2 is false
D. Statement 1 is false and Statement 2 is true.

Answer

d

Explanation

`vec(AD)=2hatj -hatk ,vec(BD) =-2hati -hatj -3hatk and vec(CD)=2hati-hatj`
volume of tetrahedron = `1/6 [vec(AD) vec(BD) vec(CD)]`
`=1/6||{:(0,2,-1),(-2,-1,-3),(2,-1,0):}||=8/3`
Also, area of the triangle ABC = `1/2 |vec(AB) xx vec(AC)|`
`1/2||{:(hati,hatj,hatk),(2,3,2),(-2,3,-1):}||`
` 1/2 |-9hati - 2hatj +12 hatk|`
`sqrt229/2`
Then ` 8/3 = 1/3 xx` (Distance of D from base ABC) `xx` (Area of triangle ABC)
Distance of D from base ABC` = 16//sqrt229`