Collapsing one Dimension

In this case, projecting w onto v involved "collapsing" the y dimension of w. Lets see if the same result comes out mathematically. The same steps work for 3D as for 2D:

1. Compute the lengths:
   • | w |  =   (keep it symbolic)
   • | v |²  =   ((5, 0, 3.1)^T · (5, 0, 3.1)^T)   =   34.61
2. Compute the unit vectors:
   • w_u  =   (4, 2.5, 2.5)^T / | w |
   • v_u  =   (5, 0, 3.1)^T / | v |
3. Compute the cosine of the angle between the vectors:
   • w_u · v_u
      =   (4, 2.5, 2.5)^T / | w | · (5, 0, 3.1)^T / | v |
      =   27.75/( | w || v |)
4. Assemble the projection:
   • kv  =   | w | (w_u · v_u) v_u
   • kv  =   | w | (27.75 / (| w || v |))  (5, 0, 3.1)^T / | v |
   • kv  =   (27.75 / (| v |))  ( (5, 0, 3.1)^T / | v | )
   • kv  =   (27.75 / | v |²)   (5, 0, 3.1)^T
   • kv  =   (27.75 / 34.61)   (5, 0, 3.1)^T
   • kv  =   ((27.75*5)/34.61, 0, (27.75*3.1)/34.61^T
   • kv  =   ( 4.00, 0, 2.49)^T

So the visual estimate kv = (4, 0, 2.5)^T matches the mathematical result, ( 4.00, 0, 2.49)^T. The orthogonal vector can easily be computed:

• u  =   w - kv
• u  =   (4, 2.5, 2.5)^T - (4, 0, 2.5)^T  =   (0, 2.5, 0)^T