(I roughly translate the problem statement from German)
Given the vectors a = (1,-2,3) and b = (1,1,1), divide the vector a in two components a1 (parallel to b) and a2 (perpendicular to b).
In a previous question of the problem, I found that:
a.b = 2
a x b = (-5, 2, 3)
|a| = √(14) ≈ 3.74
|b| = √(3) ≈ 1.73
So far, our class has been mentioning vector addition, multiplication by a real number, components of a vector, scalar product, vector product, as well as diverse properties of the operations I just listed (commutativity, distributivity, associativity and homogeneity).
The Attempt at a Solution
My problem here is that the angle between the vectors is not given in the problem. I found several ways to calculate it with the use of arccos, but since that was not yet mentioned in my class, I am reluctant to use it.
What I know so far is that:
(I'll call ∂ the angle between a and b)
a1 = |a| cos ∂
a2 = |a| sin ∂
a1 x b = 0 (because a1 and b are parallel, their cross product is equal to 0)
a2.b = 0 (because a2 and b are perpendicular, their scalar product is equal to 0)
since a.b = |a|.|b|.cos ∂, I also find that cos ∂ = 2/√(14).√(3) ≈ 0.31
I also know that a = √(a1^2 + a2^2) since the angle between a1 and a2 is π/2 rad.
What am I missing to solve this problem? I just started a program of physic after many years without maths, so I need to refresh a bit :)
Thank you very much for your answers, I appreciate it.