Finding a vector parallel to a plane

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To find a vector parallel to a plane, one must first identify the normal vector, which is perpendicular to the plane. For the plane defined by the equation 2x - 3y - z = 0, the normal vector is <2, -3, -1>. To determine a parallel vector, any vector that is not a scalar multiple of the normal vector can be used, as it will lie within the plane. Additionally, understanding that a plane can be defined by all vectors perpendicular to a fixed normal vector aids in visualizing parallel vectors. Thus, any vector that does not align with the normal vector will suffice as a parallel vector to the plane.
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Homework Statement


Just a general question for studying purposes. How do you find a vector parallel to a plane?

Let's say that we have a plane 2x-3y-z=0...I know that the perpendicular vector to this plane is &lt;2,-3,-1&gt; but how do I use this information to determine a parallel vector?
 
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Well, you know of a vector that forms a right angle, i.e., is perpendicular to, the plane, so how would you find vectors that are perpendicular to the perpendicular?

O.K, it's a mouthful. Consider a plane parallel to 2x-3y-z=0. What would be its equation? Remember the perspective of a plane as a set of vectors perpendicular
to a given fixed vector.
 

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