Find the angle between the vectors A = 2i + 2j and B = 2i - 2j. (2022)

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Question: Find the angle between the vectors A = 2i + 2j and B = 2i - 2j. (2022)

Options:

Correct Answer: 90 degrees

Exam Year: 2022

Solution:

cos(θ) = (A · B) / (|A||B|). A · B = 0, hence θ = 90 degrees.

Find the angle between the vectors A = 2i + 2j and B = 2i - 2j. (2022)

Find the angle between the vectors A = 2i + 2j and B = 2i - 2j. (2022)

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 of A and B and adding them together.
Step 3: For A = 2i + 2j, the components are (2, 2). For B = 2i - 2j, the components are (2, -2).
Step 4: Calculate A · B = (2 * 2) + (2 * -2) = 4 - 4 = 0.
Step 5: Find the magnitudes of A and B. |A| = √(2^2 + 2^2) = √(4 + 4) = √8 = 2√2. |B| = √(2^2 + (-2)^2) = √(4 + 4) = √8 = 2√2.
Step 6: Use the formula cos(θ) = (A · B) / (|A||B|). Substitute the values: cos(θ) = 0 / (2√2 * 2√2) = 0 / 8 = 0.
Step 7: Since cos(θ) = 0, this means θ = 90 degrees.

Dot Product – The dot product of two vectors is used to find the cosine of the angle between them.
Magnitude of Vectors – Calculating the magnitude of vectors is essential for determining the angle using the cosine formula.
Orthogonal Vectors – Understanding that a dot product of zero indicates that the vectors are orthogonal (90 degrees apart).