The position vectors of the points P, Q, R, S are `hat"i" + hat"j" + hat"k", 2hat"i" + 5hat"j", 3hat"i" + 2hat"j" - 3hat"k"`, and `hat"i" - 6hat"j" - hat"k"` respectively. Prove that the line PQ and RS are parallel

#### Solution

Given that the position vector of the given points P, Q, R, S are

`vec"OP" = hat"i" + hat"j" + hat"k"`

`vec"OQ" - 2hat"i" + 5hat"j"`

`vec"OR" = 3hat"i" + 2hat"j" - 3hat"k"`

`vec"OS" = hat"i" - 6hat"j" - hat"k"`

`vec"PQ" = vec"OQ" - vec"OP"`

= `(2hat"i" + 5hat"j") - (hat"i" + hat"j" + hat"k")`

= `2hat"i" + 5hat"j" - hat"i" - hat"j" - hat"k"`

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

`vec"RS" = vec"OS" - vec"OR"`

= `(hat"i" - 6hat"j" - hat"k") - (3hat"i" + 2hat"j" - 3hatk")`

= `hat"i" - 6hat"j" - hat"k" - 3hat"i" - 2hat"j" + 3hat"k"`

= `-2hat"i" - 8hat"j" + 2hat"k"`

= `-2(hat"i" + 4hat"j" - hat"k")`

`vec"RS" = -2 vec"PQ"`

∴ `vec"RS"` and `vec"PQ"` are parallel vectors.

Two vectors `vec"a"` and `vec"b"` are parallel vectors if `vec"a" = lambdavec"b"` where `lambda` is a scalar.