Question

Show that the following vectors are coplanar

`hat"i" - 2hat"j" + 3hat"k", -2hat"i" + 3hat"j" - 4hat"k", -hat"j" + 2hat"k"`

Answer 1

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.