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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.

The coordinates of the vector

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.

The vector's decomposition along the Cartesian axes

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.

The unit vectors

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.

The lengths of the projections on the axes

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 \(\alpha = 30^\circ\).

example

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 \( (8.66, 5) \).

example

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 projections of a vector in three-dimensional space

The above insights provide a fundamental understanding of vectors' behavior and decomposition, which are essential concepts in mathematics and physics.




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