Computing two vectors that are perpendicular to third vector in 3D

Question

What is the best (fastest) way to compute two vectors that are perpendicular to the third vector(X) and also perpendicular to each other?

This is ho

Answer 1

For a good HELPER vector: find the coordinate of X with the smallest absolute value, and use that coordinate axis:

absX = abs(X.x); absY = abs(X.y); absZ = abs(X.z);
if(absX < absY) {
  if(absZ < absX)
    HELPER = vector(0,0,1);
  else // absX <= absZ
    HELPER = vector(1,0,0);
} else { // absY <= absX
  if(absZ < absY)
    HELPER = vector(0,0,1);
  else // absY <= absZ
    HELPER = vector(0,1,0);
}

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Note: this is effectively very similar to @MBo's answer: taking the cross-product with the smallest coordinate axis is equivalent to setting the smallest coordinate to zero, exchanging the larger two, and negating one.