Understanding Vector Coordinates After Rotating the Coordinate System

When dealing with vector transformations in mathematics and physics, one common scenario is rotating the coordinate system while keeping the vectors stationary. This process involves understanding how the vector's coordinates change with respect to the new axes. In this article, we'll delve into the mechanics of rotating the coordinate system and how to calculate the new coordinates of a vector after such a rotation. We'll explore both conceptual and practical aspects, illustrating with examples and explanations of the math behind the transformation.

Conceptual Understanding

When you rotate the coordinate system, you're effectively reorienting the axes rather than moving the vectors themselves. This reorientation affects how vectors are perceived and represented in the new coordinate frame. The key idea is to understand how the orientation and position of the coordinate axes change and how this impacts the vector's position.

Conceptually, there are two ways to think about this rotation:

Rotation of Vectors: Imagine the vectors are pivoting around the origin as the axes are rotated. This involves a more complex transformation that typically requires knowledge of the vector in both the old and new coordinate systems. Rotation of Coordinate Axes: In this case, the vectors remain fixed, but the axes themselves are reoriented. This is the approach we'll focus on in this article because it's more straightforward mathematically and directly addresses the problem of finding the new coordinates of a vector after the axes have been rotated.

By keeping the vectors stationary and rotating the axes, we simplify the calculation of the vector's new coordinates in the transformed system.

Mathematical Approach: Using Rotation Matrices

To mathematically describe the rotation of the coordinate system, we use a rotation matrix. A rotation matrix is a square matrix that encodes the new coordinates of a point after a rotation. The identity of the rotation matrix depends on the angle and axis of rotation. For a 2D rotation, the rotation matrix can be defined as:

$$ R(theta) begin{bmatrix} cos(theta) -sin(theta) sin(theta) cos(theta) end{bmatrix} $$

where (theta) is the angle of rotation.

Given a vector (mathbf{v}) in the original coordinate system, represented as a column vector: (mathbf{v} begin{bmatrix} v_x v_y end{bmatrix}), the new coordinates (mathbf{v}') in the rotated coordinate system can be found by multiplying the original vector by the rotation matrix (R(theta)): (mathbf{v}' R(theta) mathbf{v}).

Symbolically, the new coordinates of the vector can be expressed as:

$$ mathbf{v}' begin{bmatrix} cos(theta) -sin(theta) sin(theta) cos(theta) end{bmatrix} begin{bmatrix} v_x v_y end{bmatrix} begin{bmatrix} v_x cos(theta) - v_y sin(theta) v_x sin(theta) v_y cos(theta) end{bmatrix} $$

Example: Rotating a Vector

Let's consider an example to illustrate the process. Suppose we have a vector (mathbf{v} begin{bmatrix} 3 4 end{bmatrix}) in a 2D coordinate system and we want to rotate this system by an angle of (30^circ). The rotation matrix (R(30^circ)) is:

$$ R(30^circ) begin{bmatrix} cos(30^circ) -sin(30^circ) sin(30^circ) cos(30^circ) end{bmatrix} begin{bmatrix} frac{sqrt{3}}{2} -frac{1}{2} frac{1}{2} frac{sqrt{3}}{2} end{bmatrix} $$

To find the new coordinates of the vector after the rotation, we multiply the rotation matrix by the vector:

$$ mathbf{v}' begin{bmatrix} frac{sqrt{3}}{2} -frac{1}{2} frac{1}{2} frac{sqrt{3}}{2} end{bmatrix} begin{bmatrix} 3 4 end{bmatrix} begin{bmatrix} frac{sqrt{3}}{2} cdot 3 - frac{1}{2} cdot 4 frac{1}{2} cdot 3 frac{sqrt{3}}{2} cdot 4 end{bmatrix} begin{bmatrix} frac{3sqrt{3} - 4}{2} frac{3 4sqrt{3}}{2} end{bmatrix} approx begin{bmatrix} 0.598 3.968 end{bmatrix} $$

In this example, the new coordinates of the vector after the (30^circ) rotation are approximately (begin{bmatrix} 0.598 3.968 end{bmatrix}).

Conclusion

Rotating the coordinate system and determining the new coordinates of a vector is a fundamental concept in linear algebra and has numerous applications in fields such as physics, engineering, and computer graphics. By using the rotation matrix, you can calculate the new coordinates of a vector without considering the complex transformations of the vector itself.

Understanding this process enables you to effectively manipulate geometric data in a variety of mathematical and computational contexts.

Related Keywords

Coordinate system rotation Vector transformation Rotation matrix New vector coordinates Geometry transformation