If A = 2i + 2j and B = 3i + 4j, what is the angle between A and B?

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Question: If A = 2i + 2j and B = 3i + 4j, what is the angle between A and B?

Options:

Correct Answer: 60 degrees

Solution:

cos(θ) = (A · B) / (|A||B|) = (2*3 + 2*4) / (√(2^2 + 2^2) * √(3^2 + 4^2)) = 0.5, θ = 60 degrees.

If A = 2i + 2j and B = 3i + 4j, what is the angle between A and B?

If A = 2i + 2j and B = 3i + 4j, what is the angle between A and B?

Step 1: Identify the vectors A and B. A = 2i + 2j and B = 3i + 4j.
Step 2: Calculate the dot product of A and B. A · B = (2 * 3) + (2 * 4).
Step 3: Simplify the dot product. A · B = 6 + 8 = 14.
Step 4: Calculate the magnitude of vector A. |A| = √(2^2 + 2^2) = √(4 + 4) = √8 = 2√2.
Step 5: Calculate the magnitude of vector B. |B| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
Step 6: Use the formula for the cosine of the angle θ: cos(θ) = (A · B) / (|A| * |B|).
Step 7: Substitute the values into the formula: cos(θ) = 14 / (2√2 * 5).
Step 8: Simplify the expression: cos(θ) = 14 / (10√2) = 0.5.
Step 9: Find the angle θ using the inverse cosine function: θ = cos⁻¹(0.5).
Step 10: Determine the angle: θ = 60 degrees.

Vector Operations – Understanding how to perform dot products and calculate magnitudes of vectors.
Trigonometry – Applying the cosine function to find angles between vectors.