Question

Show that the four points A, B, C and D with position vectors `4hati + 5hatj + hatk`, `-hatj-hatk`, `3hati + 9hatj + 4hatk` and `4(-hati + hatj + hatk)` respectively are coplanar

Answer 1

`bara = 4hati + 5hatj +  hatk``

`barb = -hati - hatj`

`barc = 3hati + 9hatj + 4hatk`

`bard = -4hati + 4hatj + 4hatk`

`bar(AB) = barb - bara  = - 4hati - 6hatj - 2hatk` 

`bar(AC) = barc - bara = -hati + 4hatj + 3hatj`

`bar(AD) = bard - bara = -8hati - hatj + 3hatk`

`bar(AB)`, `bar(AC)` and `bar(AD)` are coplanar if `[bar(AB) bar(AC)  bar(AD)] = 0` i.e `bar(AB).(bar(AC)xxbar(AD)) = 0`

`= [(-4,-6,-2),(-1,4,3),(-8,-1,3)]`

= -4 (12 + 3) + 6 (-3 + 24  - 2 (1+ 32)

= -4(15) + 6(21) - 2(66)

= -60+ 126 - 66

= 0

`:. bar(AB), bar(AC) ` and `bar(AD)`are coplanar

∴ Points A, B,C and D are coplanar