If A = 2i + 3j and B = 3i + 4j, what is the angle between A and B? (2021)

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Question: If A = 2i + 3j and B = 3i + 4j, what is the angle between A and B? (2021)

Options:

Correct Answer: 45 degrees

Solution:

cos(θ) = (A · B) / (|A||B|) = (2*3 + 3*4) / (√(2^2 + 3^2) * √(3^2 + 4^2)) = 0.7071, θ = 45 degrees.

If A = 2i + 3j and B = 3i + 4j, what is the angle between A and B? (2021)

If A = 2i + 3j and B = 3i + 4j, what is the angle between A and B? (2021)

Step 1: Identify the vectors A and B. A = 2i + 3j and B = 3i + 4j.
Step 2: Calculate the dot product A · B. This is done by multiplying the corresponding components: (2 * 3) + (3 * 4) = 6 + 12 = 18.
Step 3: Calculate the magnitude of vector A. |A| = √(2^2 + 3^2) = √(4 + 9) = √13.
Step 4: Calculate the magnitude of vector B. |B| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
Step 5: Use the formula for the cosine of the angle θ: cos(θ) = (A · B) / (|A||B|). Substitute the values: cos(θ) = 18 / (√13 * 5).
Step 6: Calculate the value of cos(θ). First, find √13 ≈ 3.60555, then multiply: 3.60555 * 5 ≈ 18.02775. So, cos(θ) ≈ 18 / 18.02775 ≈ 0.997.
Step 7: Use the inverse cosine function to find θ: θ = cos⁻¹(0.997). This gives θ ≈ 5.7 degrees.
Step 8: Note that the short solution states θ = 45 degrees, which seems to be incorrect based on the calculations.

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 A and B is essential for determining the angle.
Trigonometric Functions – Understanding how to use the cosine function to find angles from the dot product.