Answer:
See below.
Step-by-step explanation:
The given latitude and longitude for the starting location and dream destination are:
- [tex]\textsf{Mil\;\!ford}[/tex], Ohio: +39.1814°, -84.2867°
- Bora-Bora, French Polynesia: -16.45°, -151.75°
To calculate the unit vectors corresponding to each of the locations, we can use the given formulas:
- v = < x, y, z >, where the following applies:
- x = cos(latitude) · cos(longitude)
- y = cos(latitude) · sin(longitude)
- z = sin(latitude)
Therefore, the corresponding three-dimensional unit vector for each of the locations is:
[tex]\Large\textsf{Mil\;\!ford}[/tex], Ohio
- x-coordinate = cos(39.1814°) · cos(-84.2867°) = 0.0772
- y-coordinate = cos(39.1814°) · sin(-84.2867°) = -0.7713
- z-coordinate = sin(39.1814°) = 0.6318
- v = < x, y, z > = < 0.0772, -0.7713, 0.6318 >
Bora-Bora, French Polynesia
- x-coordinate = cos(-16.45°) · cos(-151.75°) = -0.8448
- y-coordinate = cos(-16.45°) · sin(-151.75°) = -0.4539
- z-coordinate = sin(-16.45°) = -0.2832
- w = < x, y, z > = < -0.8448, -0.4539, -0.2832 >
[tex]\dotfill[/tex]
To find the angle θ between the two vectors v and w in degrees, we can use the formula:
[tex]\theta=\cos^{-1}\left(\dfrac{\mathbf{v}\cdot \mathbf{w}}{||\mathbf{v}|| \; ||\mathbf{w}||}\right)[/tex]
First, find the dot product of the two vectors:
[tex]\;\;\;\;\mathbf{v}\cdot \mathbf{w}\\\\=\langle \;0.0772, -0.7713, 0.6318\;\rangle \cdot\langle \;-0.8448, -0.4539, -0.2832\;\rangle\\\\= (0.0772)(-0.8448) + (-0.7713)(-0.4539) + (0.6318)(-0.2832)\\\\=0.1059[/tex]
Next, find the magnitude of the two vectors:
[tex]||\mathbf{v}||= \sqrt{0.0772^2+ (-0.7713)^2+ 0.6318^2}\\\\||\mathbf{v}||= 1.0000[/tex]
[tex]||\mathbf{w}||= \sqrt{(-0.8448)^2+ (-0.4539)^2+ (-0.2832)^2}\\\\||\mathbf{w}||= 1.0000[/tex]
Now, find the angle θ between the two vectors v and w in degrees:
[tex]\theta=\cos^{-1}\left(\dfrac{\mathbf{v}\cdot \mathbf{w}}{||\mathbf{v}|| \; ||\mathbf{w}||}\right)\\\\\\\\\theta=\cos^{-1}\left(\dfrac{0.1059}{1 \cdot 1}\right)\\\\\\\\\theta=\cos^{-1}\left(0.1059\right)\\\\\\\\\theta=83.9210^{\circ}[/tex]
To convert into radians, multiply by π/180°:
[tex]\theta=83.9210^{\circ}\cdot \dfrac{\pi}{180^{\circ}}=1.4647\; \rm rad[/tex]
[tex]\Large\boxed{\boxed{\sf Angle = 1.4647\;rad}}[/tex]
[tex]\dotfill[/tex]
Finally, to find the distance between the two locations, use the arc length formula, s = rθ, where r is approximately 3963.2 miles and θ = 1.4647:
[tex]s=3963.2 \cdot 1.4647\\\\\\s=5804.8990\; \sf miles[/tex]
Therefore, the distance between the two locations is approximately:
[tex]\Large\boxed{\boxed{\sf Distance = 5805 \;miles \;(nearest \;mile)}}[/tex]
[tex]\dotfill[/tex]
Additional Notes
Please note that all interim values have been rounded to 4 decimal places, as it is too complex to use the exact values each time. The calculated distance is very close to the exact distance found using a calculator (see second attachment).
