Question

A radar station locates a sinking ship at range 15.9 km and bearing 136° clockwise from north. From the same station, a rescue plane is at horizontal range 19.6 km, 156° clockwise from north, with elevation 2.05 km. (a) Write the displacement vector from plane to ship, letting i hat represent east, j hat north, and k hat bold up.

Answer 1

Answer on Question #46316, Physics, Mechanics | Kinematics | Dynamics

A radar station locates a sinking ship at range $15.9\,\mathrm{km}$ and bearing $136{}^{\circ}$ clockwise from north. From the same station, a rescue plane is at horizontal range $19.6\,\mathrm{km}$, $156{}^{\circ}$ clockwise from north, with elevation $2.05\,\mathrm{km}$. (a) Write the displacement vector from plane to ship, letting i hat represent east, j hat north, and k hat bold up.

Solution:

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 $z$-axis. 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 $\hat{\mathbf{i}}$, $\hat{\mathbf{j}}$ and $\hat{\mathbf{k}}$ are unit vectors.

Find the xyz coordinates of each object using:

- $x =$ east

- $y =$ north

- $z =$ altitude.

For ship:

$$x_1 = 15.9 \cdot \cos(136{}^{\circ} - 90{}^{\circ}) = 15.9 \cdot \cos(46{}^{\circ}) = 11.05$$ $$y_1 = -15.9 \cdot \sin(136{}^{\circ} - 90{}^{\circ}) = -15.9 \cdot \sin(46{}^{\circ}) = -11.44$$ $$z_1 = 0$$

For rescue plane:

$$x_2 = 19.6 \cdot \cos(156{}^{\circ} - 90{}^{\circ}) = 19.6 \cdot \cos(66{}^{\circ}) = 7.972$$ $$y_2 = -19.6 \cdot \cos(156{}^{\circ} - 90{}^{\circ}) = -19.6 \cdot \sin(66{}^{\circ}) = -17.91$$ $$z_2 = 2.05$$

The displacement vector $\mathbf{d}$ from $P_1$ to $P_2$ may be written as

$$\vec{d} = (x_2 - x_1)\hat{\mathbf{i}} + (y_2 - y_1)\hat{\mathbf{j}} + (z_2 - z_1)\hat{\mathbf{k}}$$ $$\vec{d} = (7.972 - 11.05)\hat{\mathbf{i}} + (-17.91 + 11.44)\hat{\mathbf{j}} + (2.05 - 0)\hat{\mathbf{k}}$$ $$\vec{d} = -3.078\hat{\mathbf{i}} - 6.47\hat{\mathbf{j}} + 2.05\hat{\mathbf{k}}$$

Answer: $\vec{d} = -3.078\hat{\mathbf{i}} - 6.47\hat{\mathbf{j}} + 2.05\hat{\mathbf{k}}$

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