If the vectors A = (1, 2) and B = (2, 1) are given, what is the angle between them?

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Q: If the vectors A = (1, 2) and B = (2, 1) are given, what is the angle between them?

Solution: Cosine of angle θ = (A · B) / (|A| |B|) = (1*2 + 2*1) / (√5 * √5) = 4/5, θ = cos⁻¹(4/5).

Steps: 9

Step 1: Identify the vectors A and B. A = (1, 2) and B = (2, 1).
Step 2: Calculate the dot product of A and B. This is done by multiplying the corresponding components and adding them: A · B = (1 * 2) + (2 * 1).
Step 3: Simplify the dot product calculation: A · B = 2 + 2 = 4.
Step 4: Calculate the magnitude (length) of vector A. The formula is |A| = √(1^2 + 2^2) = √(1 + 4) = √5.
Step 5: Calculate the magnitude (length) of vector B. The formula is |B| = √(2^2 + 1^2) = √(4 + 1) = √5.
Step 6: Use the cosine formula to find the cosine of the angle θ: cos(θ) = (A · B) / (|A| * |B|).
Step 7: Substitute the values into the formula: cos(θ) = 4 / (√5 * √5).
Step 8: Simplify the denominator: √5 * √5 = 5, so cos(θ) = 4 / 5.
Step 9: To find the angle θ, use the inverse cosine function: θ = cos⁻¹(4/5).