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Find orthogonal vector to current vector in 3D 
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#1
Dec1712, 08:55 AM

P: 24

Hi,
In 2D I know a simple answer: vector (a,b) is orthogonal to vector (b,a) Is there anyway similar to that to find an orthogonal vector in 3D? 


#2
Dec1712, 10:36 AM

P: 685

You can use the dot product. For example, if you have a vector v and want to find vector c that is orthogonal to v, then use the dot product <v,c> and set it equal to 0.
Example: v = (4,2,3) c = (x,y,z) = ? (i) Set the dot product to zero: <v,c> = 4x + 2y + 3z = 0 (ii) Choose some values for x and y, e.g. x=0 and y=3 (iii) Solve the equation in (i) for z: z = 1/3*(4x2y) = 1/3*(0+6) = 2 Result: c = (0,3,2)  Another possibility is to use the cross product. If vector v is given, choose some vector p (not parallel to v) and form the vector c = v x p. Example: v = (4,2,3) (i) Choose an arbitrary vector p (not parallel to v): p = (0,0,1) (ii) Form the vector c = v x p (cross product): c = (4,2,3) x (0,0,1) = (2,4,0)  Note that there are infinitely many vectors that are orthogonal to a given vector. 


#3
Dec1712, 04:13 PM

P: 24

Many thanks
I knew dot and cross product, but because I write code so I need the simplest way to boost performance. As in 2D case I don't need to calculate anything, just use the trick. Also that it works for normalized vectors which doesn't need square root, a slow operation. I hope there are some tricks like that in 3D Regards 


#4
Dec1712, 04:45 PM

Engineering
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Thanks
P: 6,962

Find orthogonal vector to current vector in 3D
There isn't a unique vector orthogonal to a given vector in 3D. If the vector doesn't need to have any other properties, the same "trick" works. A vector orthogonal to (a, b, c) is (b, a, 0), or (c, 0, a) or (0, c, b).
But if you want a unit orthogonal vector, you will have to use something like a square root. 


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