[VECTORS] Got the answer, lack visual understanding

[aeromat]

Homework Statement

Which of the following lines is parallel to the plane 4x + y - z - 10 = 0?

ii)
x = -3t
y= -5 +2t
z = -10t

This is parallel, I confirmed it.

The Attempt at a Solution

I already know that this line is parallel to the plane.
I know that if that "If this line is parallel to the plane then its direction vector must be perpendicular to the plane's normal vector". But I don't see how this is true visually. I drew it out and I still don't understand why that condition must be met

 

[Pengwuino]

Are you asking for help visualizing the statement you have in quotations?

Think of something simple such as the xy-plane. Let's say you have a vector that's simply (1,0,0), so some unit vector pointing in the x-direction. Obviously it's parallel to the xy-plane right? Next you want to think about what the plane's normal vector is. Well, the xy-plane's normal vector must point in the z-direction. So your normal vector points in the z-direction (that is, a vector like (0,0,1) ), the parallel to the plane vector (1,0,0) points in the x-direction. So hopefully it's fairly obvious that those two vectors are perpendicular.

You can generalize it a little bit and say you don't need something in the x-direction alone. Any vector with no z-component will be perpendicular to that normal vector who only has a z-component.

 

[aeromat]

I understand better now, thank you. And yes, the visualization was for the post-message in quotations.

 

[HallsofIvy]

Any vector that is in the plane is certainly perpendicular to the normal vector.

And any line that is parallel to the plane is parallel to a vector in the plane.