Question

Show that the following vectors are coplanar

`2hat"i" + 3hat"j" + hat"k", hat"i" - hat"j", 7hat"i" + 3hat"j" + 2hat"k"`

Answer 1

Let the given vectors be `vec"a", vec"b"` and `vec"c"`.

`vec"a" = 2hat"i" + 3hat"j" + hat"k"`

`vec"b" = hat"i" - hat"j"`

`vec"c" = 7hat"i" + 3hat"j" + 2hat"k"`

If we are able to write `vec"a" = "m"vec"b" + "n"vec"c"`

Where m and n are scalars then we say that the vectors `vec"a", vec"b", vec"c"` are coplanar.

Let `vec"a" = "m"vec"b" + "n"vec"c"`

`2hat"i" + 3hat"j" + hat"k" = "m"(hat"i" - hat"j") + "n"(7hat"i" + 3hat"j" + 2hat"k")`

Equating `hat"i",  hat"j"` and `hat"k"` components

2 = m + 7n  .......(1)

3 = – m + 3n   .......(2)

⇒ 1 = 2n

n = `1/2`

Substituting n = `1/2` in (1) we get,

`"n" + 7(1/2)` = 2

⇒ `"m" + 7/2` = 2

⇒ m = `2 - 7/2`

m = `(4 - 7)/2`

⇒ m = `(-3)/2`

∴ m = `(-3)/2`, n = `1/2`

Substituting m and n values in (1) we get,

L.H.S = 2

R.H.S = `(-3)/2 + 7(1/2)`

= `(-3)/2 + 7/2`

= `4/2`

= 2

We are able to write `vec"a"` as a linear combination of `vec"b"` and `vec"c"`

∴ The vectors `vec"a", vec"b", vec"c"` are coplanar.