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Question
Show that the vectors `hat (i) - 2 hat(j) + 3 hat (k), - 2 hat(i) + 3 hat(j) - 4 hat(k) " and " hat(i) - 3 hat(j) + 5 hat(k) ` are coplanar.
Solution 1
`vec(a) = hat (i) - 2 hat(j) + 3 hat (k), vec(b) = - 2 hat(i) + 3 hat(j) - 4 hat(k) vec (c) hat(i) - 3 hat(j) + 5 hat(k) `
To show that these all co-planar vectors
` ⇒ vec(a) . (vec(b) xx vec (c) ) = 0 `
`vec(b) xx vec(c) = |[vec(i), vec(j) , vec(k)] , [-2 , 3 , -4],[1, -3 , 5 ]| = 3 vec(i)+ 6 vec(j) + 3vec(k)`
`vec(a) . (vec(b) xx vec(c) ) = ( vec(i) - 2 vec(j) + 3vec(k)) . (3vec(i)+ 6vec(j) + 3 vec(k))`
= 0
So all `vec( a ), vec( b ), vec( c )` all coplanar vectors
Solution 2
let `veca = hat"i" - 2hat"j" + 3hat"k"`
`vecb = -2hat"i" + 3hat"j" - 4hat"k"`
`vecc = hati - 3hat"j" + 5hat"k"`
`[veca vecb vecc] = |(1,-2,3), (-2,3,-4), (1,-3,5)|`
= 1(15 - 12) + 2(- 10 + 4) +3(6 - 3)
= 3 - 12 + 9
= 0
therefore, `veca, vecb, vecc` are coplanar
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