91Ó°ÊÓ

Cartesian coordinates represent points in a plane using two values, typically referred to as the x (horizontal) and y (vertical) coordinates. Imagine a grid where each point is determined by how far it is from the x-axis and y-axis.
For instance, if we have a point \( P(3, 4) \), it means it's 3 units to the right of the y-axis and 4 units up from the x-axis.

In the original exercise, vector \( \vec{A} \) is first given in polar coordinates with a magnitude (or length) and an angle. To work with it in equations, we convert it to Cartesian coordinates using trigonometric functions:

\( A_x = A \cos(\theta) \)

\( A_y = A \sin(\theta) \)

With \( A = 12 \text{ m} \) and \( \theta = 60^\text{°} \), this gives us: \( A_x = 6 \text{ m} \) and \( A_y = 12 \times \sin(60^\text{°}) = 6\text{ m}\times\ticksqrt{3} \), resulting in the Cartesian coordinates \( 6 \hat{\text i} + 6 \sqrt{3} \hat{\text j} \).

Polar coordinates represent points in a plane using a distance from a reference point (usually the origin) and an angle from a reference direction (usually the positive x-axis).
Imagine a clock: the center is the origin, and each point on the clock can be described by how far it is from the center (magnitude) and its angle (position around the clock).
For vector \( \vec{A} \) in polar coordinates, the magnitude is 12 meters, and the angle is 60 degrees counterclockwise from the positive x-axis. It effectively tells us how 'long' the vector is and in what direction it points.

This system is very useful for problems involving rotational movement, as you can easily describe circular paths and rotations using just a distance and an angle.

A rotation matrix is a mathematical tool used to rotate points or vectors in a plane around the origin. It helps us change the orientation of a system while keeping the same relative positions of points.

For a counterclockwise rotation by an angle \( \theta \), the rotation matrix is:
\[ \begin{bmatrix} \cos(\theta) & -\sin(\theta) \ \ \sin(\theta) & \cos(\theta) \end{bmatrix} \]
When we multiply this matrix by a vector, it rotates the vector counterclockwise by \( \theta \) degrees.
In the exercise, we rotate the vector \( \vec{A} \) by 20 degrees counterclockwise. This means using:

\[ \begin{bmatrix} \cos(20^\text{°}) & -\sin(20^\text{°}) \ \ \sin(20^\text{°}) & \cos(20^\text{°}) \end{bmatrix} \]

then multiplying that by the Cartesian coordinates of \( \vec{A} \) and \( \vec{B} \).

Trigonometry is a branch of mathematics that deals with the relationships between the angles and sides of triangles. It's especially useful for converting between polar and Cartesian coordinates.
Key trigonometric functions: \( \cos \) (cosine) and \( \sin \) (sine) help us relate the horizontal and vertical components of a vector to its magnitude and direction (angle).

In the exercise, we use these functions to convert vector \( \vec{A} \) from polar to Cartesian coordinates:

\( A_x = A \cos(60^\text{°}) = 12 \left(\tfrac{1}{2}\right) = 6 \text{ m} \)

\( A_y = A \sin(60^\text{°}) = 12 \left(\tfrac{\ticksqrt{3}}{2}\right) = 6 \sqrt{3} \)

Furthermore, when we rotate the vectors, we again rely on trigonometric functions within the rotation matrix to determine the new coordinates in the rotated system.