If the vector A = (1, 2) and B = (2, 1), what is the angle between them?

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Question: If the vector A = (1, 2) and B = (2, 1), what is the angle between them?

Options:

Correct Answer: 45 degrees

Solution:

Cosine of angle = (A · B) / (|A| |B|) = (1*2 + 2*1) / (√5 * √5) = 4/5, angle = cos^(-1)(4/5).

If the vector A = (1, 2) and B = (2, 1), what is the angle between them?

If the vector A = (1, 2) and B = (2, 1), what is the angle between them?

Step 1: Identify the vectors A and B. A = (1, 2) and B = (2, 1).
Step 2: Calculate the dot product of A and B. This is done by multiplying the corresponding components and adding them: A · B = (1 * 2) + (2 * 1) = 2 + 2 = 4.
Step 3: Calculate the magnitude (length) of vector A. |A| = √(1^2 + 2^2) = √(1 + 4) = √5.
Step 4: Calculate the magnitude (length) of vector B. |B| = √(2^2 + 1^2) = √(4 + 1) = √5.
Step 5: Use the formula for the cosine of the angle between two vectors: Cosine of angle = (A · B) / (|A| * |B|). Substitute the values: Cosine of angle = 4 / (√5 * √5) = 4 / 5.
Step 6: Find the angle by taking the inverse cosine: angle = cos^(-1)(4/5).

Dot Product – The dot product of two vectors is calculated by multiplying their corresponding components and summing the results.
Magnitude of a Vector – The magnitude of a vector is found using the formula |A| = √(x^2 + y^2) for a vector A = (x, y).
Cosine of the Angle – The cosine of the angle between two vectors can be determined using the formula cos(θ) = (A · B) / (|A| |B|).
Inverse Cosine Function – To find the angle from the cosine value, the inverse cosine function (cos^(-1)) is used.