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

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

Options:

Correct Answer: 90 degrees

Solution:

cos(θ) = (A · B) / (|A||B|) = (8) / (√8 * √8) = 1, θ = 0 degrees.

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

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

Step 1: Identify the vectors A and B. A = 2i + 2j and B = 2i - 2j.
Step 2: Calculate the dot product A · B. This is done by multiplying the corresponding components: (2 * 2) + (2 * -2) = 4 - 4 = 0.
Step 3: Calculate the magnitude of vector A. |A| = √(2^2 + 2^2) = √(4 + 4) = √8.
Step 4: Calculate the magnitude of vector B. |B| = √(2^2 + (-2)^2) = √(4 + 4) = √8.
Step 5: Use the formula for the cosine of the angle θ: cos(θ) = (A · B) / (|A||B). Substitute the values: cos(θ) = 0 / (√8 * √8) = 0.
Step 6: Find the angle θ using the inverse cosine function: θ = cos⁻¹(0). This gives θ = 90 degrees.

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