It will help to appreciate the loss of trigonometric relationships ‘hands-on’.  Hold a triangle facing you and rotate it about the horizontal x-axis in the direction of the line-of-sight (the z-axis), then repeat the process about the vertical y-axis, hold the mental picture of both rotations superimposed.  The procedure demonstrates that two normal planes rotated about perpendicular axes toward the line-of-sight, i.e. in the direction of an axis that is perpendicular to both, become as if orthonormal (to both normal axes).  (If the experiment could be carried out simultaneously with both planes they would maintain an angular relationship- that would be orthogonal- even though each has become orthonormal to both normal axes).  Next turn the triangle again facing you and rotate it about the horizontal x-axis in the direction of the line-of-sight [see animation].  Note that we cannot see its projection on any plane that lies in the line-of-sight regardless of the triangle’s (depression) angle l toward the line-of-sight, in fact the only projection we can see directly would be in the normal view.  The procedure can be repeated with the triangle turned at any angle m about the y-axis before rotating it in the direction of the line of sight to show that m and l suffice to define any angle for the plane of rotation with respect to the x- and y-axes.  It is important for our discussion that we see that the fact of vectors being perpendicular on a plane in 3-space does not necessarily mean that their projection will be so on another plane, and vice versa.  The exercise shows that for any angle q between two vectors there is a set of angles l and m on orthogonal planes (only l and one plane is shown in the accompanying animation, m is held at 0º) in the direction of the line-of-sight that will vary the projected angle between q and 180º.  In our example we find that the tangent that defines a right triangle becomes a cosine in the triangle’s projection with the ordinate becoming a hypotenuse (angle w₁₂ = 90º when cotl = cotq₁ + cotq₂).  In the limit case when l and m are both 0º, i.e. the triangle lays in the line of sight, the reflected angle w₁₂ is 180º regardless of the angles q₁ and q₂.  This loss of correlation is pivotal to our discourse: any vector in an imaginary plane regardless of its direction will be seen as orthonormal with respect to a normal (real) plane and conversely a vector that appear as orthonormal may have any direction in an imaginary plane.