What is the angle between the vectors a = (1, 2, 2) and b = (2, 0, 2)?
Correct Answer: 45 degrees
- Step 1: Identify the vectors a and b. Here, a = (1, 2, 2) and b = (2, 0, 2).
- Step 2: Calculate the dot product a · b. This is done by multiplying corresponding components and adding them together: 1*2 + 2*0 + 2*2.
- Step 3: Perform the calculations for the dot product: 1*2 = 2, 2*0 = 0, and 2*2 = 4. Now add these results: 2 + 0 + 4 = 6.
- Step 4: Calculate the magnitude of vector a, |a|. This is done using the formula |a| = √(1^2 + 2^2 + 2^2).
- Step 5: Perform the calculations for |a|: 1^2 = 1, 2^2 = 4, and 2^2 = 4. Now add these: 1 + 4 + 4 = 9, then take the square root: |a| = √9 = 3.
- Step 6: Calculate the magnitude of vector b, |b|. Use the formula |b| = √(2^2 + 0^2 + 2^2).
- Step 7: Perform the calculations for |b|: 2^2 = 4, 0^2 = 0, and 2^2 = 4. Now add these: 4 + 0 + 4 = 8, then take the square root: |b| = √8 = 2√2.
- Step 8: Use the formula for the cosine of the angle θ: cos(θ) = (a · b) / (|a| |b|). Substitute the values: cos(θ) = 6 / (3 * 2√2).
- Step 9: Calculate the denominator: 3 * 2√2 = 6√2. Now substitute this into the equation: cos(θ) = 6 / (6√2).
- Step 10: Simplify the fraction: cos(θ) = 1 / √2.
- Step 11: Find the angle θ by taking the inverse cosine: θ = cos⁻¹(1/√2).
- Step 12: Determine the angle in degrees: θ = 45 degrees.
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