Find parametric equations for the line with the following properties. The line passes through the origin, it is contained in the plane $x-2y+z=0$, and is orthogonal to the vector $v = \langle3,4,2\rangle$.
I know that the vector is orthogonal to $v$ when the dot product of the vector I am looking for and $v$ is equal to $0$. But, how would I make sure that said vector is also meeting the other requirements?
Best Answer
The parametric equations will be of the form
$$x=0+at $$ $$y=0+bt $$ $$z=0+ct$$ with
$$3a+4b+2c=0$$ and $$at-2bt+ct=0$$
thus
$$a=2b-c $$ and
$$6b-3c+4b+2c=0$$ which gives $$c=10b $$ and $$a=-8b $$ finally, we get
$$x=-8t $$ $$y=t $$ $$z=10t $$