For vectors : planes,
one of the specific instructional objectives states this " interpret and find equations of planes in the cartesian form, scalar product form, and parametric form , and convert the equations from one form to another "
Qn : how to convert the abovementioned forms into parametric form?? How to split a normal vector into two vectors perpendicular to the normal vectors (other than obnoxiously difficult guess and check method)?
A normal vector can split into infinite sets of two perpendicular vectors.
Just remember that u⊥v ⇔ u·v = 0 .
Example: Given u = (1, 3, 4), I can easily think of vectors that are perpendicular. I simply set vâ‚ƒ = 0, then think: What should I set vâ‚� and vâ‚‚ to be, in order that
uâ‚�vâ‚� + uâ‚‚vâ‚‚ + uâ‚ƒvâ‚ƒ = 0?
Simple, just pick vâ‚� = -3 and vâ‚‚ = 1. So v = (-3, 1, 0) is ⊥ to u.
We can similarly and easily also show that the following two vectors are also ⊥ to u:
w = (-4, 0, 1), t = (0, -4, 3).
Please see my free H2 Mathematics Textbook for more.