Question

Find the vector equation of the line passing through the point having position vector `2hati + hatj - 3hatk` and perpendicular to vectors `hati + hatj + hatk and hati + 2hatj - hatk`.

Answer 1

Let `barb = hati + hatj + hatk and barc = hati + 2hatj - hatk`

The vector perpendicular to the vectors and `barb` and `barc` is given by

`barb xx barc = |(hati, hatj, hatk),(1, 1, 1),(1, 2, -1)|`

= `hati (-1-2) - hatj(-1-1) + hatk(2 - 1)`

= `-3hati + 2hatj + hatk`

Since the line is perpendicular to the vectors `barb` and `barc`, it is parallel to `barb xx barc`. 

The vectors equation of the line passing through `A(bara)` and parallel to `barb xx barc` is `barr = bara + lambda (barb xx barc)`, where λ is a scalar.

Here, `bara = 2hati + hatj - 3hatk`

Hence, the vector equation of the required line is `barr = (2hati + hatj - 3hatk) + lambda(-3hati + 2hatj + hatk)`.