Question

If `hati + hatj + hatk, hati - hatj + hatk` and `2hati + 3hatj + mhatk` are coplanar then find the value of m.

Answer 1

Let `bara = hati + hatj + hatk`,

`barb = hati - hatj + hatk`,

`barc = 2hati + 3hatj + mhatk`

Since `bara, barb, barc` are coplanar we have:

`bara = xbarb + ybarc`, where x, y are scalars.

∴ `hati + hatj + hatk = x(hati - hatj + hatk) + y(2hati + 3hatj + mhatk)`

= `(x + 2y)hati + (-x + 3y)hatj + (x + my)hatk`

Equating the coefficients of `hati, hatj, hatk`, we get

x + 2y = 1    ...(1)

–x + 3y = 1  ...(2)

x + my = 1   ...(3)

Solving (1) and (2):

x = `1/5`,

y = `2/5`

From (3) `1/5 + m(2/5)` = 1

∴  `(2m)/5 = 1 - 1/5 = 4/5`

∴ 10m = 20

∴  m = 2