Final answer:
The question relates to the properties of parallelograms and vector algebra. Without a diagram or additional context, it is not possible to accurately verify if |AQ| = 2 |AB| or determine the distance from point P to line AB. The given excerpts from the text offer some insights into vector operations and geometry, but do not provide sufficient information to solve the specific questions presented.
Step-by-step explanation:
The given problem pertains to the properties of a parallelogram and vectors. In part (a), we look to determine if |AQ| = 2 |AB| is true. The information provided contains vector algebra concepts and geometric constructions used to find the resultant and difference of two vectors by creating parallelograms. Unfortunately, the provided information is not sufficient to conclusively show |AQ| = 2 |AB| without additional context or a diagram. Typically, one might use vector addition to find that in a parallelogram, the diagonals bisect each other, which could imply |AQ| = 2 |AB| if point Q is the midpoint of diagonal BD. However, a correct answer can't be given with the current details.
In part (b), the question asks us to show that a point P lies on line AC and determine the distance from P to line AB. The provided information suggests using the baseline AB and line AC to find the parallax and apply proportions, however, without a clear setup or visual it's challenging to determine the correct answer. A more detailed geometric approach or further data points would be needed to solve this accurately.
Finally, while the problem mentions rotations and invariance, the concepts described are generally related to vector magnitudes and scalar multiplication, yet these details do not offer a clear path to resolving the questions in part (b) on the distance from P to line AB.