i) Find the vector that represents the flight path of Airplane 1. Show your steps.

Question

Answer the questions based on the given information. 

The flight path of two airplanes in a flight simulator game are shown below. The coordinates of the airports P and Q are given.

Airplane 1 flies directly from P to Q.

 Airplane 2 has a layover at R and then flies to Q.

The path of Airplane 2 from P to R can be represented by the vector \(5\hat {i}+\hat {j}-2\hat {k}\).

(Note: Assume that the flight path is straight and fuel is consumed uniformly throughout the flight.)

i) Find the vector that represents the flight path of Airplane 1. Show your steps. 

ii) Write the vector representing the path of Airplane 2 from R to Q. Show your steps.

iii) What is the angle between the flight paths of Airplane 1 and Airplane 2 just after takeoff? 

OR

iii) Consider that Airplane 1 started the flight with a full fuel tank. 

Find the position vector of the point where a third of the fuel runs out if the entire fuel is required for the flight.

Answer 1

i) Writes the vectors for points P and Q as follows:

Finds the vector representing the flight path of Airplane 1 as:

ii) Uses vector subtraction to find the vector representing the flight path from R to Q as:

iii) Finds the cosine of the angle between the vectors representing the flight paths of Airplane 1 and Airplane 2 as:

Finds the angle between the flight paths as:

\(\theta =cos ^ {-1}(\frac {18}{5\sqrt {15}})\)

OR 

iii) Considers a point S which divides PQ internally in the ratio 1:2. 

Finds the position vector of point S as: