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, we will use the dot product formula:
u.v = |u||v|cos(θ)
First, let's calculate the dot product of u and v:
u.v = (0.15342 * -0.37833) + (0.88745 * 0.58981) + (0.43463 * 0.71344)
u.v = -0.05799 + 0.52345 + 0.31013
u.v = 0.77559
Next, let's find the magnitudes of vectors u and v:
|u| = sqrt(0.15342^2 + 0.88745^2 + 0.43463^2)
|u| = sqrt(0.0235604564 + 0.7888444025 + 0.1886968769)
|u| = sqrt(1.0011017347)
|u| ≈ 1.00055
|v| = sqrt((-0.37833)^2 + 0.58981^2 + 0.71344^2)
|v| = sqrt(0.1431275889 + 0.3477890361 + 0.5094271136)
|v| = sqrt(1.0003437386)
|v| ≈ 1.00017
Now, we can substitute the dot product and magnitudes back into the dot product formula to find the angle θ:
0.77559 = 1.00055 * 1.00017 * cos(θ)
0.77559 = 1.0007203 * cos(θ)
cos(θ) = 0.77559 / 1.0007203
cos(θ) ≈ 0.77462
Now we need to find the angle whose cosine is approximately 0.77462. We can use arccos to find θ:
θ = arccos(0.77462)
θ ≈ 0.70129 radians
Therefore, the angle θ between vectors u and v is approximately 0.70129 radians.
how did you convert to radians
To convert between degrees and radians, you can use the fact that π radians is equivalent to 180 degrees.
To convert from radians to degrees:
- Multiply the angle in radians by 180/π
To convert from degrees to radians:
- Multiply the angle in degrees by π/180
In the previous response, I calculated the angle in radians θ = 0.70129. To convert this angle to degrees, you can use the formula:
θ (degrees) = θ (radians) * 180/π
θ (degrees) = 0.70129 * 180/π
θ (degrees) ≈ 40.20692 degrees
Therefore, the angle θ between vectors u and v is approximately 40.20692 degrees.
but in the first one you said it was 0.70129 radians.
Apologies for the confusion in my previous responses. Let's correct this.
The angle θ between vectors u and v was previously calculated to be approximately θ ≈ 0.70129 radians. To ensure consistency, let's keep the answer in radians.
Therefore, the correct angle between vectors u and v is approximately θ ≈ 0.70129 radians.