Find the value of m for which the line `vecr.=(hati+2hatj-hatk)+lambda(2hati+hatj+2hatk)` is parallel to the plane `vecr.(3hati-2hatj+mhatk)=12`

Explanation

We know that the line `vecr=veca+lambdavecb` is parallel to the plane
`vecr.vecn=q` only when `vecb.vecn=0`.
Here, `vecb=(2hati+hatj+2hatk)` and `vecn=(3hati-2hatj+mhatk)`.
So, the given line is parallel to the given plane
`ltimplies vecb.vecn=0`
`ltimplies (2hati+hatj+2hatk).(3hati-2hatj+mhatk)=0`
`ltimplies (2 xx 3) + 1 xx (-2) + 2 xx m=0`
`ltimplies 2m=-4 ltimplies m=-2`. ltbr Hence, the required value of m is `-2`.