Transformation matrix, vector algebra word problem

[mathclass]

Hi everyone. I am not sure if this problem belongs under the "Linear & Abstract algebra" section but it seemed like it may. Please let me know if there is a different section that would better fit this problem.

So here is a word problem that is proposed:
A solar panel is capable of rotating independently about a fixed x,y,z axis. The solar panel start by pointing upward in the z direction and the center of the pannel is located at (0,0,0). At a certain time of day the sun relative to the x y z origin is in the direction of the vector (1,1,1). Calculate the transformation matrix which can be applied to the original solar panel such that the maximum power is obtained (normal to panel is aligned with sun direction). Note that the center is kept at (0,0,0).

If anyone has an idea how to set up and solve this problem please post. Thank you for your help!

 

[Unco]

Hi everyone. I am not sure if this problem belongs under the "Linear & Abstract algebra" section but it seemed like it may. Please let me know if there is a different section that would better fit this problem.

Yep, it's linear algebra.

So here is a word problem that is proposed:
A solar panel is capable of rotating independently about a fixed x,y,z axis. The solar panel start by pointing upward in the z direction and the center of the pannel is located at (0,0,0). At a certain time of day the sun relative to the x y z origin is in the direction of the vector (1,1,1). Calculate the transformation matrix which can be applied to the original solar panel such that the maximum power is obtained (normal to panel is aligned with sun direction). Note that the center is kept at (0,0,0).

If anyone has an idea how to set up and solve this problem please post. Thank you for your help!

By "The solar panel start by pointing upward in the z direction", do you mean the normal of the plane at the origin is in the z-direction?

If so, the solar panel is in the x-y plane and is the spanned by the standard unit vectors e1 = (1,0,0) and e2 = (0,1,0).

The (desired) plane through the origin perpendicular to the vector (1,1,1) is equal to the span of {(-1, 1, 0), (-1, 0, 1)} --- found by solving (x, y, z).(1, 1, 1)=0 --- so a transformation that maps e1 to (-1,1,0) and e2 to (-1, 0, 1) will suffice. A matrix representation with respect to the standard basis {e1, e2, e3} follows.

Let us know if you have problems, showing your work!

 

[mathclass]

Thank you for the reply Unco. I understand how the span of those vectors were found but I am not totally sure about how to set up the actual transformation matrix. I am somewhat new to transformation matrices and just want to make sure it is done properly. Could you please help me with writing the transformation matrix?

Thank you for all the help, I really appreciate it.

 

[Unco]

I understand how the span of those vectors were found but I am not totally sure about how to set up the actual transformation matrix. I am somewhat new to transformation matrices and just want to make sure it is done properly. Could you please help me with writing the transformation matrix?

If A is a linear map that takes basis elements of R3 (say) b1, b2, b3 to c1, c2, c3, respectively, then, by definition, the matrix representation of A with respect to the basis {b1, b2, b3} is given by [c1 c2 c3].

 

[mathclass]

Sorry but I still do not know how I am suppose to set up the transformation matrix.
Also wouldn't we want to use the cross product not the dot product?