
The Cartesian Decomposition of a Vector
Cartesian decomposition is a process that divides a vector into its constituent parts along the axes of the Cartesian coordinate system. In mathematical terms, it's expressed as $$ \vec{v} = x \cdot \vec{u}_x + y \cdot \vec{u}_y + z \cdot \vec{u}_z $$
When we represent a vector on a Cartesian plane, we can decompose it into its Cartesian components—two numerical values that represent its projections onto the \(x\) and \(y\) axes.
This method of decomposition is a crucial tool in physics and mathematics, allowing us to analyze vectors in terms of their coordinates.
It proves particularly useful when dealing with two- or three-dimensional spaces.
A Practical Example
Consider a vector v (4;5) within a two-dimensional space.
$$ \vec{v} = \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 4 \\ 5 \end{pmatrix} $$
Here, the vector's starting point is at the origin (0;0), and its endpoint is at the coordinates (4;5).
You can visualize this vector on a Cartesian plane.
By projecting the second endpoint onto the Cartesian axes, you obtain two new vectors: vx on the x-axis and vy on the y-axis.
It's easy to see that the sum of vx+vy equals the original vector v.
This can be confirmed through the parallelogram method or the head-to-tail method.
$$ \vec{v} = \vec{v}_x + \vec{v}_y $$
The original vector v can be expressed as the sum of its components, considering vx=4·ux and vy=5·uy
$$ \vec{v} = 4 \cdot \vec{u}_x + 5 \cdot \vec{u}_y $$
Here, ux and uy are unit vectors, or vectors with a length of one, along the Cartesian axes.
In general, any vector can be decomposed in this manner.
$$ \vec{v} = x \cdot \vec{u}_x + y \cdot \vec{u}_y $$
Where "x" and "y" are the magnitudes of the vectors vx and vy that have been projected onto the Cartesian axes.
Finding the Components of a Vector
You can easily calculate the "x" and "y" components of a vector using some simple trigonometry.
All you need is the vector’s magnitude, represented as $ | \vec{v} | $, and the angle \(\alpha\), measured counterclockwise from the \(x\)-axis.</p>
$$ x = | \vec{v} | \cdot \cos \alpha = 4 $$
$$ y = | \vec{v} | \cdot \sin \alpha = 5 $$
In this case, |v| is the magnitude of the initial vector, and alpha is its angle relative to the positive x-axis.
These trigonometric identities are extremely useful in physics and are worth remembering.
Here’s a practical example.
Imagine a vector with a magnitude of \( | \vec{v} | = 10 \) units and an angle \(\alpha = 30^\circ\).
The \(x\)-component is approximately 8.66:
$$ x = 10 \cdot \cos 30^\circ = 10 \cdot \frac{\sqrt{3}}{2} \approx 8.66 $$
The \(y\)-component comes out to 5:
$$ y = 10 \cdot \sin 30^\circ = 10 \cdot \frac{1}{2} = 5 $$
So, the vector’s Cartesian coordinates are \( (8.66, 5) \).
A vector can also be decomposed in three-dimensional space: $$ \vec{v} = x \cdot \vec{u}_x + y \cdot \vec{u}_y + z \cdot \vec{u}_z $$ Here, the Cartesian decomposition is represented by three components (vx, vy, vz), corresponding to projections along the x, y, and z axes, respectively.
The above insights provide a fundamental understanding of vectors' behavior and decomposition, which are essential concepts in mathematics and physics.