Find the angle θ between your 2 vectors. Show your work and final answer. Make sure your final angle θ is in radians not degrees. Use 5 decimal places in all your calculations.
u=0.15342,0.88745,0.43463
v=-0.37833,0.58981,0.71344
Then
Find the distance between the two locations to the nearest mile. Show your work and final answer with units. (2 points). Remember your angle needs to be in radians when you compute arc length.
Although Earth is not a perfect sphere, assume that it is for the purposes of this portfolio and use the formula for the arc length of a segment of a great circle on a sphere, s = rθ, to find the distance between your two locations. In this case, r is the radius of Earth, which is approximately 3,963.2 miles. Find the distance between your two locations.
To find the angle θ between the two vectors u and v, we can use the dot product formula:
u • v = |u||v|cos(θ)
First, calculate the magnitudes of the vectors:
|u| = sqrt(0.15342^2 + 0.88745^2 + 0.43463^2) = 1.02284
|v| = sqrt(-0.37833^2 + 0.58981^2 + 0.71344^2) = 1.00000
Next, calculate the dot product:
u • v = 0.15342(-0.37833) + 0.88745(0.58981) + 0.43463(0.71344) = 0.27904
Now, plug the values into the dot product formula to find θ:
0.27904 = 1.02284 * 1.00000 * cos(θ)
cos(θ) = 0.27904 / (1.02284 * 1.00000)
cos(θ) ≈ 0.27306
θ ≈ arccos(0.27306) = 1.28515 radians
Now, to find the distance between the two locations, we use the formula for arc length:
s = rθ
s = 3963.2 * 1.28515
s ≈ 5091.814 miles
Therefore, the distance between the two locations is approximately 5091.814 miles.