1. A vector u is given by u = λa + μb, where λ, μ are elements of ℝ. Calculate axu and bxu and show that:
[itex][(b χ u).n]/[(b χ a).n][/itex] = λ
[itex][(a χ u).n]/[(a χ b).n][/itex] = μ
(In any circumstance when I have omitted the underlining it is for convenience purposes, a, b, n and u are still vectors.)
2. The vectors a and b form a set of basis vectors for P1 (A plane spanned by vectors a and b). However it is often desirable to use a set of orthogonal basis vectors.
Construct a vector b' that is in P 1 and orthogonal to a.
We are told that a and b are vectors in ℝ3, and that they are not parallel or anti-parallel. n is the normal vector to plane P1.
The Attempt at a Solution
1. n= a x b
a x b = a x (aλ x bμ)
= a x bμ
∴[itex](a χ u)/n[/itex] = μ
[itex](a χ u)/(a χ b)[/itex] = μ
The same can be applied for obtaining an equation with λ. The problem arises out of the fact that I wouldn't know how to incorporate the '.n' portion to the numerator and denominator (though it seems quite obvious that their cancellation would yield the above equation, it must serve some purpose).
2. I can't find any sort of methodology that would yield a vector, b', perpendicular to a in P1 when only given the information above. I would assume it has something to do with question 1 (I could be wrong), though I can't see how they relate to one another.
Thanks in advance for any help.