a. With the help of a sketch show that in polar computation given two-dimensionalcoordinates (N1,E1) of a point P1,and the distance L and bearing α of another point P2from P1,that the coordinates (N2,E2) of P2 are given by; (6 Marks)N2= N1 + L Cos α and E2= E1 + L Sin α
Question
a. With the help of a sketch show that in polar computation given two-dimensionalcoordinates (N1,E1) of a point P1,and the distance L and bearing α of another point P2from P1,that the coordinates (N2,E2) of P2 are given by; (6 Marks)N2= N1 + L Cos α and E2= E1 + L Sin α
Solution
To illustrate this, let's start with a simple sketch:
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Draw a Cartesian coordinate system with North (N) on the y-axis and East (E) on the x-axis.
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Plot a point P1 at coordinates (N1, E1).
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From point P1, draw a line at an angle α from the North direction. This line represents the bearing of point P2 from P1.
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The length of this line is L, which is the distance from P1 to P2.
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At the end of this line, plot the point P2.
Now, let's derive the coordinates of P2:
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The change in the North direction (y-axis) from P1 to P2 is the length of the line L times the cosine of the angle α. This is because cosine of an angle in a right triangle is defined as the ratio of the adjacent side (North direction in this case) to the hypotenuse (L). So, the North coordinate of P2 is N1 (North coordinate of P1) plus L cos α. Hence, N2 = N1 + L cos α.
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Similarly, the change in the East direction (x-axis) from P1 to P2 is the length of the line L times the sine of the angle α. This is because sine of an angle in a right triangle is defined as the ratio of the opposite side (East direction in this case) to the hypotenuse (L). So, the East coordinate of P2 is E1 (East coordinate of P1) plus L sin α. Hence, E2 = E1 + L sin α.
This shows that the coordinates of P2 are given by N2 = N1 + L cos α and E2 = E1 + L sin α.
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