Given the parameters p_1,p_2,p_3 does the following system can be solved: This system actually discribes a Rotation Matrix p_1 a bc p_2 de f p_3 where only p_1, p_2 and _p3 are know. For example p_1=p_2=p_3=1 would result in a=b=c=d=e=f=0. Why is this unabigiously) solvable / not solvable? If it is solvable, what is the solution?

Alfredeim 2022-09-15 Answered
Given the parameters p 1 , p 2 , p 3 does the following system can be solved:
This system actually discribes a Rotation Matrix
p 1     a         b c         p 2     d e         f         p 3
where only p 1 , p 2 and p 3 are know. For example p 1 = p 2 = p 3 = 1 would result in a = b = c = d = e = f = 0.
Why is this system (unabigiously) solvable / not solvable?
If it is solvable, what is the solution?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Dalton Erickson
Answered 2022-09-16 Author has 10 answers
Let
M = ( p 1 a b c p 2 d e f p 3 ) .
If M is a rotation matrix, then M 1 = M T . This implies that M and M 1 have the same entries on the main diagonal. But if M is not the identity, M M 1 , so the rotation matrix is not completely determined by the entries on its main diagonal.
On the other hand, if [ u 1   u 2   u 3 ] T is a unit vector on the axis of rotation of M and if 𝜃 is the angle of rotation about that axis, then
(1) p i = ( 1 cos θ ) u i 2 + cos θ
for i = 1 , 2 , 3. Moreover,
p 1 + p 2 + p 3 = 1 + 2 cos θ .
Therefore we can express cos𝜃 in terms of p 1 ., p 2 ., and p 3 .. Plug that value of cos𝜃 into equation (1) for each i ; this either gives u 1 = 0 or gives two possible values of u 1 which differ only by a sign change.
We can safely assume that 0 θ π ,, because the rotation described by angle θ and unit vector [ u 1   u 2   u 3 ] T is the same as the rotation described by angle θ and unit vector [ u 1   u 2   u 3 ] T .. That means that in general there are eight possible ways to fill in the matrix M (one for each choice of the signs of each of the u i ) ,), therefore eight possible solutions to the given set of equations. (For 0 < θ < π ,, there are four solutions if exactly one of the u i is zero, two solutions if two of the 𝑢𝑖 are equal to zero. There are half as many solutions if θ = π ,, and of course only one solution if θ = 0. ).) Moreover, by computing the rotation matrix for the rotation by angle θ around the axis given by [ u 1   u 2   u 3 ] T ,, we can compute all the unknown entries a , b , c , d , e , and f in the rotation matrix for a specific choice of [ u 1   u 2   u 3 ] T .
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2020-10-25

What are the solutions to the following system of equations?
{y=x2+3x73xy=2

asked 2022-07-06
1.) Given the following equations:
3 x y = 30 5 x 3 y = 10
What are the values of x and y?
asked 2022-06-09
solve the system for x, y and z. Is there any smart trick to solve it?
{ 2 a ( a x + b y ) + 2 c ( c x + d y ) + 2 z x = 0 2 b ( a x + b y ) + 2 d ( c x + d y ) + 2 z y = 0 x 2 + y 2 1 = 0
a , b , c , d R and ( x , y , z ) R 3 .
asked 2022-08-03
Solve the following system of equations. If there are no solutions, type "No Solution" for both x and y. If there are infinitely many solutions, type "x" for x, and an expression in terms of x for y.
1 x + 3 y = 8
2 x + 6 y = 16
x =
y =
asked 2022-06-21
Find all complex values for a where there's no solution for the non homogeneous system
{ x + ( 1 a ) y + z = 1 2 x y + z = 3 3 x a y + ( a 1 ) z = 4
After moving to a matrix representation and reduction, I have:
( 1 1 a 1 | 1 0 2 a 3 1 | 1 0 0 a 1 | 0 )
Now, I don't see an a that answering this request, and I double-checked my reduction process.
asked 2022-04-04

Solve for the variables b and c in the system of equations:
{6b+2c=26b+4c=3

asked 2022-07-12
How to solve this this equations?
555 = 0.862 X + 0.138 Y 0.345 Z , 911 = 0.413 X + 0.587 Y 0.492 Z , 674 = 0.849 X + 0.142 Y + 0.358 Z ,