Show that the following vectors are coplanar
`hat"i" - 2hat"j" + 3hat"k", -2hat"i" + 3hat"j" - 4hat"k", -hat"j" + 2hat"k"`
Solution
Let the given vectors be `vec"a" = hat"i" - 2hat"j" + 3hat"k"`
`vec"b" = -2hat"i" + 3hat"j" - 4hat"k"`
and `vec"c" = -hat"j" + 2hat"k"`
Three vectors `vec"a", vec"b"` and `vec"c"` are coplanar if one vector is expressed as a linear combination of the other two vectors.
`vec"a" = "s"vec"b" + "t"vec"c"` where s, t are scalars.
Let `vec"a" = "s"vec"b" + "t"vec"c"` where s, t are scalars.
`hat"i" - 2hat"j" + 3hat"k" = "s"(-2hat"i" + 3hat"j" - 4hat"k") + "t"(-hat"j" + 2hat"k")`
`hat"i" - 2hat"j" + 3hat"k" = -2"s"hat"i" + (3"s" - "t")hat"j" + (-4"s" + 2"t")hat"k"`
1 = – 2S ........(1)
– 2 = 3s – t .......(2)
3 = – 4s + 2t
(1) ⇒ s = `-1/2`
Substituting for s in equation (2) we get
– 2 = `3 xx - 1/2 - "t"`
– 2 = `- 3/2 - "t"`
t = `- 3/2 + 2`
= `(- 3 + 4)/2`
t = `1/2`
Substituting for s and t in equation (3)
(3) ⇒ 3 = `-4 xx -1/2 + 2 xx 1/2`
3 = + 2 + 1
3 = 3
∴ Equation (3) is satisfied.
The scalars s and t exist.
∴ The vectors `vec"a" = hat"i" - 2hat"j" + 3hat"k", vec"b" = -2hat"i" + 3hat"j" - 4hat"k"` and `vec"c" = -hat"j" + 2hat"k"` are coplanar vectors.
