Vectors Parallel to plane

[markraz]

Hi,

If I am only given a normal (green) and a point (orange), is it possible to find 2 vectors (red and blue) parallel to the normal as shown in the pic? I know I can find a plane equation from a normal and a point. But can I find the blue and red vectors from just a point and a normal? I read somewhere you can use the cross product of 2 vectors to find another vector perpendicular. Can you do something like that here in reverse?

thank you in advance.

 

[Deleted member 4993]

Hi,

If I am only given a normal (green) and a point (orange), is it possible to find 2 vectors (red and blue) parallel to the normal as shown in the pic? I know I can find a plane equation from a normal and a point. But can I find the blue and red vectors from just a point and a normal? I read somewhere you can use the cross product of 2 vectors to find another vector perpendicular. Can you do something like that here in reverse?

thank you in advance.

You can find the plane on which those two vectors lie - but you cannot "uniquely" define those "red" and "blue" vectors, without some other relevant information. You cannot tell which two numbers multiplied made up 24.

 

[markraz]

You can find the plane on which those two vectors lie - but you cannot "uniquely" define those "red" and "blue" vectors, without some other relevant information. You cannot tell which two numbers multiplied made up 24.

Thanks, can you give me some idea what the other required info would be? Do I need to know some additional points or additional vectors?
Thanks

 

[Ishuda]

Thanks, can you give me some idea what the other required info would be? Do I need to know some additional points or additional vectors?
Thanks

The red and blue vector plane can be translated and rotated to the z=0 plane with the green vector either straight up or straight down from the origin at (0,0,0). This is the usual two dimensional plane we plot our y=f(x) on and it will be sufficient to talk of vectors (directed line segments) in this plane with the understood z always equals zero. So the red vector can be though of as part of the line
R: y = m x
Now if m were zero, that line would be y=0 and the normal to that would be the line x=0. But, if m were not zero, how would the line normal to R through the point (0,0) be defined?

Having answered that question, you would then have two normal lines but in what direction do you want the vectors to point. Again using the example m=0, the red vector could be pointed in the positive y direction or the negative y direction. The same type of thing would apply to the blue vector.

 

[markraz]

The red and blue vector plane can be translated and rotated to the z=0 plane with the green vector either straight up or straight down from the origin at (0,0,0). This is the usual two dimensional plane we plot our y=f(x) on and it will be sufficient to talk of vectors (directed line segments) in this plane with the understood z always equals zero. So the red vector can be though of as part of the line
R: y = m x
Now if m were zero, that line would be y=0 and the normal to that would be the line x=0. But, if m were not zero, how would the line normal to R through the point (0,0) be defined?

Having answered that question, you would then have two normal lines but in what direction do you want the vectors to point. Again using the example m=0, the red vector could be pointed in the positive y direction or the negative y direction. The same type of thing would apply to the blue vector.

thanks appreciate it. rotating the plane to zero seems logical, I never would have though of that.. now the question is.. how do you rotate the plane to zero? in the z?

if the equation of the plane equation is in general form ax+by+cz+constant = 0... can this easily be manipulated to be rotated to zero? in the Z?? by some standard procedure? I looked it up in my book just now but I couldn't find an example

thanks again

 

[Ishuda]

thanks appreciate it. rotating the plane to zero seems logical, I never would have though of that.. now the question is.. how do you rotate the plane to zero? in the z?

if the equation of the plane equation is in general form ax+by+cz+constant = 0... can this easily be manipulated to be rotated to zero? in the Z?? by some standard procedure? I looked it up in my book just now but I couldn't find an example

thanks again

Suppose you have a plane given by
a₀ x + b₀ y + c₀ z + d₀ = 0
If we rotate two of the axis, say x and y, by choosing an angle theta and letting
u = C x + S v
v = -S x + C y
where C and S are the cosine and sine of the angle theta respectively and, at the same time, assuming c₀ is not zero, do a translation on z letting
w = z + d₀/c₀
we now have
x = C u - S v
y = S u + C v
z = w - d₀/c₀
This leads to
a₀ (C u - S v) + b₀ (S y + C v) + c₀ ( w - d₀/c₀) + d₀
= (a₀ C + b₀ S) u + (-a₀ S + b₀ C) v + c₀ w = 0

Let
a₁ = (a₀ C + b₀ S)
b₁ = c₀
and choose the angle theta so that
\(\displaystyle \theta\, =\, tan^{-1}(\frac{b_0}{a_0})\)
so that
-a₀ S + b₀ C = 0
and we have a translated rotated plane
a₁ u + b₁ w = 0

Now do another rotation. Note that since all we did was rotations and translations, lengths were preserved. That is
(u₂-u₁)² + (v₂-v₁)² + (w₂-w₁)² = (x₂-x₁)² + (y₂-y₁)² + (z₂-z₁)²