What is the angle between the vectors (1, 2, 2) and (2, 1, 2)?

What is the angle between the vectors (1, 2, 2) and (2, 1, 2)?

Step 1: Identify the vectors. We have vector u = (1, 2, 2) and vector v = (2, 1, 2).
Step 2: Calculate the dot product of the vectors u and v. The dot product is calculated as u · v = (1*2) + (2*1) + (2*2).
Step 3: Perform the multiplication: (1*2) = 2, (2*1) = 2, and (2*2) = 4. Now add them together: 2 + 2 + 4 = 8. So, u · v = 8.
Step 4: Calculate the magnitude of vector u. The magnitude |u| is calculated as √(1^2 + 2^2 + 2^2).
Step 5: Perform the calculations: 1^2 = 1, 2^2 = 4, and 2^2 = 4. Now add them: 1 + 4 + 4 = 9. So, |u| = √9 = 3.
Step 6: Calculate the magnitude of vector v. The magnitude |v| is calculated as √(2^2 + 1^2 + 2^2).
Step 7: Perform the calculations: 2^2 = 4, 1^2 = 1, and 2^2 = 4. Now add them: 4 + 1 + 4 = 9. So, |v| = √9 = 3.
Step 8: Use the cosine formula to find the cosine of the angle θ: cos(θ) = (u · v) / (|u| |v|).
Step 9: Substitute the values: cos(θ) = 8 / (3 * 3) = 8 / 9.
Step 10: Use the inverse cosine function to find θ: θ = cos⁻¹(8/9).
Step 11: Calculate θ to find that it is approximately 60 degrees.

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