91Ó°ÊÓ

To understand coordinate rotation, imagine a standard Cartesian coordinate system where a vector is defined by its components along the x and y axes. Now, if we rotate this coordinate system by an angle \( \theta \), the components of the vector in the new coordinate system change their orientation but maintain their magnitude.
One way to think about this is by visualizing the x and y axes as a spinning wheel: as this wheel turns, the projections of a vector onto these axes shift accordingly.
The transformation of vector components due to such a rotation is mathematically described by the rotation formulas:

• \( x' = x \cos \theta + y \sin \theta \)
• \( y' = -x \sin \theta + y \cos \theta \)

These equations show how vector components \( x \) and \( y \) transform into \( x' \) and \( y' \) based on the rotation angle \( \theta \). In our exercise, the angle \( \theta \) is \( 60^{\circ} \), reflecting typical transformations that can occur in physical or geometric contexts.

The components of a vector describe the vector's magnitude along specific directions. Here, we deal with two basic components:

• \( x \)-component
• \( y \)-component

Each of these components corresponds to values in a coordinate system that define the vector’s orientation.
In the given exercise, the vector’s original components are \((n+1)\) along the x-axis and \(1\) along the y-axis. Upon rotating the coordinate system by \( 60^{\circ} \), these components change values, transitioning to \( n \) for the x-direction and \( 3 \) for the y-direction.
The goal is to determine \( n \), recognizing that vector components are independent of the coordinate system but dependent on their projection onto the axes. Solving the associated equations helps bridge the transition from one set of components to another, as the vector maintains the same physical reality across transformations.

Trigonometric functions are pivotal in transforming vector components during coordinate rotation. These functions include sine and cosine, which are directly tied to angles in triangle geometry.
In the context of our exercise, the rotation angle is \( 60^{\circ} \), with known trigonometric values:

• \( \cos 60^{\circ} = \frac{1}{2} \)
• \( \sin 60^{\circ} = \frac{\sqrt{3}}{2} \)

The sine function (\( \sin \theta \)) assesses the ratio of the opposite side over the hypotenuse in a right triangle, while cosine (\( \cos \theta \)) considers the adjacent side over the hypotenuse. Using these in vector transformation enables precise calculation of new vector components after rotation.
In practical applications, understanding these relationships allows for the precise execution of geometric operations or in calculating forces in physics, enhancing our ability to manipulate and understand vectorial properties amid varying lines of action.