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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
Solution
`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
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