How is the x-component of a vector calculated?

The x-component of a vector in a polar coordinate system can be determined using the relationship between the vector's magnitude, its angle, and the cosine function. Specifically, if the vector has a magnitude (r) and is oriented at an angle (\theta) from the positive x-axis, the x-component can be calculated by multiplying the length of the vector (r) by the cosine of the angle (\theta). This reflects the horizontal projection of the vector onto the x-axis.

The formula used is (x = r \cdot \cos(\theta)), which captures how the angle influences the vector's reach along the x-axis. This foundational concept in physics and mathematics allows for the breakdown of vectors into their respective components, which is crucial for analyzing forces, motion, and other vector quantities.

In this context, the other formulas do not accurately represent the x-component of a vector. Using sine instead of cosine would instead yield the y-component. Dividing (r) by cosine does not correctly represent any vector component and does not adhere to the principles of vector decomposition. Adding (r) to the cosine of the angle does not provide any meaningful vector relationship and does not represent the geometrical interpretation that aligns with