If the vectors `4hati+11hatj+mhatk,7hati+2hatj+6hatk and hati+5hatj+4hatk` are coplanar, then m is equal to

Options

A. 38
B. 0
C. 10
D. `-10`

Answer

C

Explanation

Since the three vectors are coplanar, one will be a linear combination off the other two.
`therefore4hati+11hatj+mhatk=x(7hati+2hatj+6hatk)+y(hati+5hatj+4hatk)`
`implies4=7x+y` . . . (i)
`11=2x+5y` . . . (ii)
`m=6x+4y` . . . (iii)
From eqs. (i) and (ii), we get
`x=(3)/(11) and y=(23)/(11)`
From eq. (iii), we get
`m=6xx(3)/(11)+4xx(23)/(11)=10`
Trick since, vectors `4hati+11hatj+mhatk,7hati+2hatj+6hatk and hati+5hatj+4hatk` are complanar.
`therefore|(4,11,m),(7,2,6),(1,5,6)|=0`
`implies4(8-30)-11(28-6)+m(35-2)=0`
`implies-88-11xx22+33m=0`
`implies-8-22+3m=0`
`implies3m=30impliesm=10`