What is the angle between the vectors A = i + j and B = 2i + 2j?

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Question: What is the angle between the vectors A = i + j and B = 2i + 2j?

Options:

Correct Answer: 45 degrees

Solution:

cos(θ) = (A · B) / (|A||B|) = (1*2 + 1*2) / (√2 * √8) = 4 / (2√2) = √2. θ = 45 degrees.

What is the angle between the vectors A = i + j and B = 2i + 2j?

What is the angle between the vectors A = i + j and B = 2i + 2j?

Step 1: Identify the vectors A and B. A = i + j and B = 2i + 2j.
Step 2: Calculate the dot product of A and B. A · B = (1 * 2) + (1 * 2) = 2 + 2 = 4.
Step 3: Calculate the magnitude of vector A. |A| = √(1^2 + 1^2) = √(1 + 1) = √2.
Step 4: Calculate the magnitude of vector B. |B| = √(2^2 + 2^2) = √(4 + 4) = √8 = 2√2.
Step 5: Use the formula for the cosine of the angle θ: cos(θ) = (A · B) / (|A||B|).
Step 6: Substitute the values into the formula: cos(θ) = 4 / (√2 * 2√2).
Step 7: Simplify the denominator: √2 * 2√2 = 2 * 2 = 4. So, cos(θ) = 4 / 4 = 1.
Step 8: Find θ by taking the inverse cosine: θ = cos⁻¹(1).
Step 9: Since cos(0) = 1, θ = 0 degrees.

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