Find 3x3 Rotation Matrix for Rotated x`y`z` Space

In summary, to find a 3x3 rotation matrix that takes a point in regular cartesian space and gives its coordinates in a rotated x'y'z' space, one can use the unit basis vectors of x'y'z' space to form a 3x3 matrix. In this case, the unit basis vectors for x', y', and z' were found to be [1/sqrt(2), 0, 1/sqrt(2)], [1/sqrt(6), -2/sqrt(6), -1/sqrt(6)], and [1/sqrt(2), -1/sqrt(6), -1/sqrt(3)], respectively. However, it was discovered that the two bottom rows
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Frillth
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Homework Statement



I need to find a 3x3 rotation matrix that takes a point in regular cartesian space and gives its coordinates in a rotate x`y`z` space. The +z` axis runs along the vector [1,1,-1], and the +x` axis should be in the xz plane with positive x component.

Homework Equations



The rotation matrix can be found by taking the unit basis vectors of x'y'z' space and putting them side by side to form a 3x3 matrix.

The Attempt at a Solution



First, I found the unit basis vector of z', which is [1/sqrt(3), 1/sqrt(3), -1/sqrt(3)]. Now I know that since x' has a basis vector in the xz plane, it has y component 0. setting x` (dot) z` = 0 gives that x` must have unit basis vector [1/sqrt(2), 0, 1/sqrt(2)]. Now I can find y' by doing z` (cross) x`, which gives the unit basis vector for y` to be: [1/sqrt(6), -2/sqrt(6), -1/sqrt(6)].

Now when I put these into matrix form, I get:

[{1/Sqrt[2], 1/Sqrt[6], 1/Sqrt[3]}, {0, -2/Sqrt[6],
1/Sqrt[3]}, {1/Sqrt[2], -1/Sqrt[6], -1/Sqrt[3]}]

However, if I take this times a vector such as [1, 1, -1] (which should come out to [0, 0, sqrt(3)] in the primed axes), I don't get the correct answer. What am I doing wrong?

Edit: Ugh, sorry. I figured it out. These completely obvious things sometimes slip past me when I'm really tired.
 
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  • #2
The two bottom rows were swapped. It should be:[{1/Sqrt[2], 0, 1/Sqrt[3]}, {1/Sqrt[6], -2/Sqrt[6], 1/Sqrt[3]}, {1/Sqrt[2], -1/Sqrt[6], -1/Sqrt[3]}]
 

FAQ: Find 3x3 Rotation Matrix for Rotated x`y`z` Space

What is a rotation matrix?

A rotation matrix is a mathematical tool used to describe the rotation of a coordinate system around a specific point or axis. It is a 3x3 matrix that represents the transformation of points from one coordinate system to another through a rotation.

How is a rotation matrix used in 3D space?

In 3D space, a rotation matrix is used to describe the orientation of an object or coordinate system in three dimensions. It can be used to rotate points, vectors, or entire coordinate systems around a specific point or axis.

How do you find a 3x3 rotation matrix for a rotated x`y`z` space?

To find a 3x3 rotation matrix for a rotated x`y`z` space, you need to know the angle and axis of rotation. This information can then be used to construct the rotation matrix using mathematical formulas, such as the Euler-Rodrigues formula or the axis-angle representation.

What is the purpose of a 3x3 rotation matrix?

The purpose of a 3x3 rotation matrix is to describe the orientation of an object or coordinate system in three dimensions. It is a useful tool in mathematics and science, particularly in fields such as computer graphics, robotics, and physics.

Can a 3x3 rotation matrix be used for non-orthogonal rotations?

Yes, a 3x3 rotation matrix can be used for non-orthogonal rotations. However, it is more commonly used for orthogonal rotations, where the axes are perpendicular to each other. For non-orthogonal rotations, other mathematical tools may be used, such as quaternions or rotation matrices of higher dimensions.

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