# Transformation matrix, vector algebra word problem

• mathclass
In summary, the problem involves finding the transformation matrix for a solar panel that can rotate independently and is initially pointing upward in the z direction, with its center located at (0,0,0). The goal is to align the panel's normal with the direction of the sun, given by the vector (1,1,1) relative to the origin. The transformation can be found by mapping the standard unit vectors e1 and e2 to (-1,1,0) and (-1,0,1) respectively, and the transformation matrix can be represented by [(-1,1,0) (-1,0,1)]. The use of the cross product may also be necessary.
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!

mathclass said:
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.

mathclass said:
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!

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.

mathclass said:
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].

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?

## 1. What is a transformation matrix?

A transformation matrix is a mathematical representation of a transformation that can be applied to a vector or set of coordinates. It is used to translate, rotate, scale, or shear objects in a coordinate system.

## 2. How do you multiply a vector by a transformation matrix?

To multiply a vector by a transformation matrix, the vector is written as a column matrix and multiplied by the transformation matrix using matrix multiplication. The resulting matrix is then converted back into a vector.

## 3. Can a transformation matrix be used to rotate a vector in 3D space?

Yes, a transformation matrix can be used to rotate a vector in 3D space. The transformation matrix will have 3 rows and 3 columns, and the rotation will be defined by the values in the matrix.

## 4. How can vector algebra be applied to solve a word problem?

Vector algebra can be used to represent and manipulate quantities that have both magnitude and direction. By setting up equations and using vector operations, word problems involving motion, forces, and other physical quantities can be solved.

## 5. What is the importance of understanding transformation matrices and vector algebra for a scientist?

Understanding transformation matrices and vector algebra is important for a scientist because it allows for the manipulation and analysis of data in multiple dimensions. This is essential for fields such as physics, engineering, and computer science, where complex systems and equations are often represented using vectors and matrices.

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