Final answer:
The distance between points in a coordinate system is calculated using the Pythagoras theorem and remains constant under coordinate rotations. Displacement is a vector quantity indicating the direction and magnitude from the start to the end point. Converting polar to Cartesian coordinates is done via cosine and sine functions.
Step-by-step explanation:
To determine the distance between two points P and Q in a coordinate system, we use the distance formula derived from the Pythagoras theorem, which in two dimensions is √((x2 - x1)² + (y2 - y1)²). In a three-dimensional space, the formula becomes √((x2 - x1)² + (y2 - y1)² + (z2 - z1)²). This formula shows that the distance is invariant under rotations of the coordinate system because the squares of the differences in coordinates remain the same even if the axis are rotated. Displacement, on the other hand, is the vector that points from the initial to the final position and has both magnitude and direction, making it a vector quantity.
To convert from polar to Cartesian coordinates, we use the formulas x = r cos(θ) and y = r sin(θ). The distance of point P to the origin can be shown to be invariant under rotations by noting that the sum of squares of the coordinates (x² + y² + z²) remains constant, which represents the distance squared in three-dimensional space.