Find the angle between the vectors A = i + j and B = 2i + 2j.

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Question: Find the angle between the vectors A = i + j and B = 2i + 2j.

Options:

Correct Answer: 0 degrees

Solution:

cos(θ) = (A · B) / (|A| |B|). A · B = 1*2 + 1*2 = 4; |A| = √2, |B| = 2√2. Thus, cos(θ) = 4 / (√2 * 2√2) = 1, θ = 0 degrees.

Find the angle between the vectors A = i + j and B = 2i + 2j.

Find 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 A · B. This is done by multiplying the corresponding components: (1 * 2) + (1 * 2) = 2 + 2 = 4.
Step 3: Calculate the magnitude of vector A. The magnitude |A| is calculated as √(1^2 + 1^2) = √(1 + 1) = √2.
Step 4: Calculate the magnitude of vector B. The magnitude |B| is calculated as √(2^2 + 2^2) = √(4 + 4) = √8 = 2√2.
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) = 4 / (√2 * 2√2).
Step 6: Simplify the denominator: |A| * |B| = √2 * 2√2 = 2 * 2 = 4.
Step 7: Substitute back into the equation: cos(θ) = 4 / 4 = 1.
Step 8: Find the angle θ. Since cos(θ) = 1, θ = 0 degrees.

Dot Product – The dot product of two vectors is calculated as the sum of the products of their corresponding components.
Magnitude of Vectors – The magnitude of a vector is calculated using the formula |A| = √(x^2 + y^2) for a 2D vector.
Angle Between Vectors – The angle between two vectors can be found using the cosine of the angle derived from the dot product and magnitudes.