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
To find the angle between two vectors, you can use the dot product formula:
u . v = |u| |v| cos(θ)
First, calculate the dot product of u and v:
u . v = (0.15342) * (-0.37833) + (0.88745) * (0.58981) + (0.43463) * (0.71344)
u . v = -0.0580454 + 0.5239723 + 0.3100865
u . v = 0.7760134
Next, calculate the magnitudes of vectors u and v:
|u| = sqrt(0.15342^2 + 0.88745^2 + 0.43463^2)
|u| = sqrt(0.0235567 + 0.7883765 + 0.1888356)
|u| = sqrt(1.0007689)
|u| = 1.000384
|v| = sqrt((-0.37833)^2 + 0.58981^2 + 0.71344^2)
|v| = sqrt(0.1432274 + 0.3474975 + 0.5094563)
|v| = sqrt(1.0001812)
|v| = 1.0000905
Now substitute these values back into the dot product formula:
0.7760134 = 1.000384 * 1.0000905 * cos(θ)
0.7760134 = 1.0004752 * cos(θ)
cos(θ) = 0.7760134 / 1.0004752
cos(θ) = 0.77585513
To find the angle θ in radians, take the arccos of 0.77585513:
θ = arccos(0.77585513)
θ ≈ 0.71952 radians
Therefore, the angle θ between vectors u and v is approximately 0.71952 radians.
0.77547 = 1.00064 * cos(θ)
To find the angle θ in radians when 0.77547 = 1.00064 * cos(θ), follow these steps:
Solve for cos(θ):
cos(θ) = 0.77547 / 1.00064
cos(θ) ≈ 0.774979
Find the angle θ in radians:
θ = arccos(0.774979)
θ ≈ 0.72202 radians
Therefore, the angle θ between the two vectors is approximately 0.72202 radians.
use this formula
u∙v=‖u‖‖v‖cos θ
cos θ=u∙v/‖u‖‖v‖
θ=cos-1u∙v/‖u‖‖v‖
to solve
Given the formula u∙v = ‖u‖‖v‖cos(θ), we can calculate the angle θ between vectors u and v by following these steps:
1. Calculate the dot product u∙v:
u∙v = (0.15342)(-0.37833) + (0.88745)(0.58981) + (0.43463)(0.71344)
u∙v = -0.0580458 + 0.5239736 + 0.3100908
u∙v = 0.7760186
2. Calculate the magnitudes of vectors u and v:
‖u‖ = sqrt(0.15342^2 + 0.88745^2 + 0.43463^2) = sqrt(0.0235566 + 0.7883802 + 0.1888473) = sqrt(1.0007841) = 1.00039
‖v‖ = sqrt((-0.37833)^2 + 0.58981^2 + 0.71344^2) = sqrt(0.1432328 + 0.3474890 + 0.5094643) = sqrt(1.0001861) = 1.00009
3. Substitute these values back into the formula to find cos(θ):
cos(θ) = 0.7760186 / (1.00039 * 1.00009) = 0.7760186 / 1.00048 = 0.7757911
4. Calculate the angle θ in radians:
θ = cos^(-1)(0.7757911) ≈ 0.72329 radians
Therefore, the angle θ between vectors u and v is approximately 0.72329 radians.
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.