Question

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

Answer 1

Let `bara = hati + hatj + hatk`,

`barb = 2hati - hatj + hatk`

Since the required line is perpendicular to both the vectors `bara` and `barb`, it is parallel to `bara xx barb`

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

= `2hati + hatj - 3hatk`

Now the required line passes through `hati + 2hatj + 3hatk` and parallel to `2hati + hatj - 3hatk`

∴ The vector equation is given by `barr = (hati + 2hatj + 3hatk) + λ(2hati + hatj - 3hatk)`