Show that the points whose position vectors `4hat"i" + 5hat"j" - hat"k", - hat"j" - hat"k", 3hat"i" + 9hat"j" + 4hat"k"` and `-4hat"i" + 4hat"j" + 4hat"k"` are coplanar

#### Solution

Let the given position vectors of the points A, B, C, D be

`vec"OA" = 4hat"i" + 5hat"j" - hat"k"`

`vec"OB" = - hat"j" - hat"k"`

`vec"OC" = 3hat"i" + 9hat"j" + 4hat"k"`

`vec"OD" = -4hat"i" + 4hat"j" + 4hat"k"`

`vec"AB" = vec"OB" - vec"OA"`

= `(-hat"j" - hat"k") - (4hat"i" + 5hat"j" + hat"k")`

= `-hat"j" - hat"k" - 4hat"i" - 5hat"j" - hat"k"`

`vec"AB" = -4hat"i" - 6hat"j" - 2hat"k"`

`vec"BC" = vec"OC" - vec"OB"`

= `(3hat"i" + 9hat"j" + 4hat"k") - (-hat"j" - hat"k")`

= `3hat"i" + 9hat"j" + 4hat"k" + hat"j" + hat"k"`

`vec"BC" = 3hat"j" + 10hat"j" + 5hat"k"`

`vec"CD" = vec"OD" - vec"OC"`

= `(-4hat"i" + 4hat"j" + 4hat"k") - (3hat"i" + 9hat"j" + 4hat"k")`

= `-4hat"i" + 4hat"j" + 4hat"k" - 3hat"i" - 9hat"j" - 4hat"k"`

`vec"CD" = -7hat"i" - 5hat"j"`

To prove the point A, B, C, D to be coplanar, it is enough to prove the vectors `vec"AB", vec"BC", vec"CD"` are coplanar.

To prove `vec"AB", vec"BC", vec"CD"` are coplanar

It is enough to prove `vec"AB" = "s"vec"BC" + "t"vec"CD"`

Three vectors `vec"AB", vec"BC", vec"CD"` are coplanar

If one vector is written as a linear combination of other two vectors.

`vec"AB" = "s"vec"BC" - "t"vec"CD"`.

∴ `(-4hat"i" - 6hat"j" - 2hat"k") = "s"(3hat"i" + 10hat"j" + 5hat"k") + "t"(-7hat"i" - 5hat"j")`

`-4hat"i" - 6hat"j" - 2hat"k" = (3"s" - 7"t")hat"i" + (10"s" - 5"t")hat"j" + 5"s"hat"k"`

Equating the like terms on both sides

– 4 = 3s – 7t ........(1)

– 6 = 10s – 5t ........(2)

– 2 = 5s ........(3)

(3) ⇒ s = `- 2/5`

Substituting in equation (2), we have

– 6 = 10 × `-2/5 - 5"t"`

– 6 = – 4 – 5t

– 6 + 4 = – 5t

⇒ – 5t = – 2

⇒ t = `2/5`

Substituting for s and t in equation (1), we have

(1) ⇒ – 4 = `3 xx -2/5 - 7 xx 2/5`

– 4 = `(-6)/5 - 14/5`

– 4 = `(-20)/5`

– 4 = – 4

∴ The sclars s and t exist.

`vec"AB" = "s"vec"BC" + "t"vec"CD"`

Hence `vec"AB", vec"BC", vec"CD"` are coplanar.

Therefore, the points A, B, C, D are coplanar points.