PYTHON$-$CALCULUS $2$ PROJECT

Solve all the question by using computer programming and verify all computation by hand calculation. Provide an explanation for each programming code you create, assuming you are explaining it to someone who doesn't understand.

Problem $2$

Vectors and Geometry of Space

In a high$-$tech exploration team, Captain Li and her crew are preparing for an expedition to a newly discovered moon. Their navigation system requires precise calculations involving coordinates and vectors to identify safe landing zones. Two potential landing sites have been identified, represented by vectors. The first landing site $P$ is at the coordinates $P(1,-2,4)$ and the second landing site $Q$ is at $Q(3,1,6).$

a$)$ Captain Li needs to establish a safe operational area. She determines that the operational zone can be defined as a parallelogram formed by the vectors vec$(PQ)$ and a third vector vec$(D)=($:$1,0,-1$:$).$

$[15$ points$]$ Calculate the vector vec$(PQ)$ that represents the direction from point $P$topo${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}Q,$ and compute the area of the parallelogram formed by vectors vec$(PQ)$ and vec$(D).$

b$)$ To ensure operational safety, the team's drone is stationed at point $R(2,0,1).$ They need to verify that the drone is within a safe distance from the landing plane defined by the equation $x+2y-z=3.$

$[10$ points$]$ Proof that point R is not on the given plane and calculate the distance from point $R$ to the landing plane.