I'm having problems with what might be a simple vectors problem.
Give an example of a vector equation of a line that is:
(a) parallel to the $xy$-plane.
(b) perpendicular to the $yz$-plane.
(c) perpendicular to the $y$-axis in $\mathbb{R}^{3}$, and passing through the origin.
Now for (a) I know that the $xy$-plane is $z=0$ and that for (b) the $yz$-plane is $x=0$ but I'm not sure how to incorporate that information into a vector equation.
I was thinking something like $\vec{v}= s(1,0,0)+t(0,1,0)$ for (a) but I feel like I'm going in the wrong direction. I'm at a complete loss for (b) and (c).
Thanks!
Best Answer
Your idea with a) is almost right, but you take a plane instead of a line and the special case, where your plane is the XY-plane itself. In general, a line $\vec{v}=\vec{p}+s\cdot (x,y,0)^T$
is parallel to the XY-plane for all points $\vec{p}$ and all values $x,\;y$ (without $x=y=0$).
In b) we get $\vec{v}=\vec{p}+s\cdot (1,0,0)^T$
for all points $\vec{p}$ since the vector $(1,0,0)^T$ is the normal vector of the YZ-plane.
In c) we get $\vec{v}=s\cdot (x,0,z)^T$
for all values $x,\;z$ (without $x=z=0$) since this line passing the origin for $s=0$ and is perpendicular to the y-axis.