Question

A radar station locates a sinking ship at range 19.5 km and bearing 136° clockwise from north. From the same station, a rescue plane is at horizontal range 19.6 km, 166° clockwise from north, with elevation 1.95 km.

(a) Write the displacement vector from plane to ship, letting î represent east, ĵ north, and k up.

(b) How far apart are the plane and ship? 2 km

Answer 1

A convenient way to specify the position of an object is with the help of a coordinate
system. We choose a fixed point, called the origin and three directed lines, which pass through
the origin and are perpendicular to each other. These lines are called the coordinate axes of a
three-dimensional rectangular (Cartesian) coordinate system and are labeled the x-, y-, and zaxis.
Three numbers with units specify the position of a point P. These numbers are the x-, y-, and
z-coordinates of the point P. Here i^, j^and k^ are unit vectors.
Find the xyz coordinates of each object using:
+x = east
+y = north
+z = altitude.
For ship:
x1 = 19.5 · cos(136° - 90°) = 19.5 · cos(46°) = 13.54
y1 = -19.5 · sin(136° - 90°) = -19.5 · sin(46°) = -14.02
z1 = 0
For rescue plane:
x2 = 19.6 · cos(166° - 90°) = 19.6 · cos(76°) = 4.74
y2 = -19.6 · cos(166° - 90°) = -19.6 · sin(76°) = -19.01
z2 = 1.95

The displacement vector d from P1 to P2 may be written as
d = (x2 - x1)i + (y2 - y1)j+ (z2 - z1)k

d = (4.74 - 13.54)i + (-19.01 + 14.02)j + (1.95 - 0)k
d = -8.8?? - 4.99?? + 1.95k====================================================(a)

(b)distance = sqrt(8.8^2 + 4.99^2 + 2.0^2) = 10.31km