Matrix for transforming vector components under rotation

In summary, the conversation discusses the use of a matrix to transform vector components from an unprimed basis to a rotated primed basis. It is mentioned that the matrix equation for this transformation involves the dot product of the unit basis vectors in the primed and unprimed coordinate systems. The conversation then moves on to a specific example in which the matrix that transforms the original components to the rotated components is found to be the inverse of the matrix used in the previous equation. The speaker is unsure where they went wrong and asks for clarification.
  • #1
saadhusayn
22
1
Say we have a matrix [itex]L[/itex] that maps vector components from an unprimed basis to a rotated primed basis according to the rule [itex]x'_{i} = L_{ij} x_{j}[/itex]. [itex]x'_i[/itex] is the [itex]i[/itex]th component in the primed basis and [itex]x_{j}[/itex] the [itex]j[/itex] th component in the original unprimed basis. Now [itex]x'_{i} = \overline{e}'_i. \overline{x} = \overline{e}'_i. \overline{e}_j x_{j} [/itex]. Hence [itex]L_{ij} = \overline{e}'_i. \overline{e}_j[/itex]. Thus the matrix equation relating the primed co-ordinate system to the unprimed one in [itex]\mathbb{R}^3[/itex] is

$$ \begin{pmatrix}x'_{1}\\ x'_{2}\\ x'_{3} \end{pmatrix} = \begin{pmatrix} \overline{e}'_1. \overline{e}_1 & \overline{e}'_1. \overline{e}_2 & \overline{e}'_1. \overline{e}_3\\ \overline{e}'_2. \overline{e}_1 & \overline{e}'_2. \overline{e}_2 & \overline{e}'_2. \overline{e}_1 \\ \overline{e}'_3. \overline{e}_1 & \overline{e}'_3. \overline{e}_2 & \overline{e}'_3. \overline{e}_3 \end{pmatrix}\begin{pmatrix}x_{1}\\ x_{2}\\ x_{3} \end{pmatrix} $$

Where the [itex]\overline{e}'_i[/itex]s and [itex]\overline{e}_j[/itex]s are unit basis vectors in the primed and unprimed co ordinate systems respectively.

Now I tried to apply the above idea to the following situation (Riley, Hobson and Bence Chapter 26, Problem 2).

I took [itex]\mathbf{A} = \overline{e}_1[/itex] and [itex]\mathbf{B} = \overline{e}_2[/itex]. It turns out that the matrix that transforms [itex]\mathbf{A} \rightarrow \mathbf{A}' [/itex] and [itex]\mathbf{B} \rightarrow \mathbf{B}' [/itex] is not the matrix that transforms the unprimed components to the primed components (that I used above) but the INVERSE (or transpose) of that matrix. I need to know where I am going wrong here. Thank you.
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  • #2
I am not sure I understand your question, but if you want to go in the opposite direction, you use the inverse matrix. Was that the question?
 
  • #3
WWGD said:
I am not sure I understand your question, but if you want to go in the opposite direction, you use the inverse matrix. Was that the question?

In the original frame,
[tex]\mathbf{A} = \begin{pmatrix} 1 \\
0 \\
0
\end{pmatrix}[/tex]

and in the rotated frame, the components of [itex] \mathbf{A}[/itex] are given by [tex] \mathbf{A'} = \begin{pmatrix} \frac{\sqrt{3}}{2} \\
0 \\
\frac{1}{2}
\end{pmatrix}[/tex]

and

in the original frame, [tex]\mathbf{B} = \begin{pmatrix} 0 \\
1 \\
0
\end{pmatrix}[/tex]

and in the rotated frame, the components of [itex] \mathbf{B}[/itex] are given by [tex] \mathbf{B'} = \begin{pmatrix} -\frac{{1}}{2} \\
0 \\
\frac{\sqrt{3}}{2}
\end{pmatrix}[/tex]

I want to find the matrix that gives you the components of any vector in the rotated frame if you have the components in the original frame. In other words, I want the matrix [itex] \mathbf{L}[/itex] such that [itex] \mathbf{x'} = \mathbf{L} \mathbf{x}[/itex]. Now from tensor analysis, we know that the [itex] L_{ij} = e'_i.e_j [/itex] where [itex]e'_i [/itex] = ith basis vector in rotated frame and [itex]e_j [/itex] = jth basis vector in original frame. I let [itex] e_1 = \mathbf{A} [/itex], [itex] e'_1 = \mathbf{A'}[/itex], [itex]e_2 = \mathbf{B}[/itex] and [itex]e'_2 = \mathbf{B'}[/itex]
And in the original frame, [tex]\mathbf{e_3} = \begin{pmatrix}
0 \\
0 \\
1
\end{pmatrix}[/tex]

And in the rotated frame,

[tex]\mathbf{e_3}' = \begin{pmatrix}
a \\
b \\
c
\end{pmatrix}[/tex]I used the fact that [itex]\mathbf{L}[/itex] has a determinant of 1 and that it has orthonormal rows to find [itex]a,b [/itex] and [itex] c [/itex].

[tex] \mathbf{L} = \begin{pmatrix} \overline{e}'_1. \overline{e}_1 & \overline{e}'_1. \overline{e}_2 & \overline{e}'_1. \overline{e}_3\\ \overline{e}'_2. \overline{e}_1 & \overline{e}'_2. \overline{e}_2 & \overline{e}'_2. \overline{e}_1 \\ \overline{e}'_3. \overline{e}_1 & \overline{e}'_3. \overline{e}_2 & \overline{e}'_3. \overline{e}_3 \end{pmatrix} = \frac{1}{2} \begin{pmatrix} \sqrt{3} & 0 & 1 \\ -1 & 0 & \sqrt{3}\\ 0 & -1 & 0\end{pmatrix} [/tex]

This, however, is not the correct matrix. It is the INVERSE of the correct matrix. Where am I going wrong?
 
Last edited:

1. What is a matrix for transforming vector components under rotation?

A matrix for transforming vector components under rotation is a mathematical tool used to rotate a vector in a three-dimensional space. It allows for the conversion of the vector's components from one coordinate system to another, taking into account the angle of rotation.

2. How does the matrix for transforming vector components under rotation work?

The matrix for transforming vector components under rotation uses a combination of trigonometric functions and geometric principles to calculate the new coordinates of the vector after rotation. It involves multiplying the original vector's components by a rotation matrix, which is a specific type of transformation matrix.

3. What is the importance of the matrix for transforming vector components under rotation?

The matrix for transforming vector components under rotation is crucial in many fields of science and engineering. It is used in computer graphics, robotics, physics, and other applications where the rotation of objects is involved. It allows for accurate and efficient calculations of rotated vectors.

4. How is the matrix for transforming vector components under rotation derived?

The matrix for transforming vector components under rotation is derived from the principles of linear algebra and geometry. It involves understanding the properties of rotation matrices and how they affect the coordinates of a vector. It also requires knowledge of trigonometry and basic algebraic operations.

5. Are there different types of matrices for transforming vector components under rotation?

Yes, there are several types of matrices for transforming vector components under rotation, depending on the specific rotation being performed. These include 2D rotation matrices, 3D rotation matrices, and matrices for specific types of rotations such as Euler angles or quaternions. Each type has its own set of properties and uses.

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