Find the angle between the vectors A = (1, 2, 2) and B = (2, 1, 1).

Find the angle between the vectors A = (1, 2, 2) and B = (2, 1, 1).

Correct Answer: 45°

Step 1: Identify the vectors A and B. A = (1, 2, 2) and B = (2, 1, 1).
Step 2: Calculate the dot product A · B. This is done by multiplying the corresponding components of A and B and then adding them together: A · B = (1*2) + (2*1) + (2*1) = 2 + 2 + 2 = 6.
Step 3: Calculate the magnitude of vector A, denoted as |A|. Use the formula |A| = √(1^2 + 2^2 + 2^2). This gives |A| = √(1 + 4 + 4) = √9 = 3.
Step 4: Calculate the magnitude of vector B, denoted as |B|. Use the formula |B| = √(2^2 + 1^2 + 1^2). This gives |B| = √(4 + 1 + 1) = √6.
Step 5: Use the formula for the cosine of the angle θ between the vectors: cos(θ) = (A · B) / (|A| |B|). Substitute the values: cos(θ) = 6 / (3 * √6).
Step 6: Simplify the expression: cos(θ) = 6 / (3√6) = 2 / √6.
Step 7: To find the angle θ, take the inverse cosine: θ = cos⁻¹(2 / √6).
Step 8: Calculate θ using a calculator or trigonometric tables to find that θ is approximately 45°.

Dot Product – Understanding how to calculate the dot product of two vectors.
Magnitude of Vectors – Calculating the magnitude (length) of a vector using the formula √(x^2 + y^2 + z^2).
Cosine of the Angle – Using the cosine formula to find the angle between two vectors.