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

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Question: If vector A = 2i + 3j and vector B = 3i + 4j, what is the angle between them? (2023)

Options:

Correct Answer: 60 degrees

Exam Year: 2023

Solution:

cos(θ) = (A · B) / (|A| |B|) = (2*3 + 3*4) / (√(2^2 + 3^2) * √(3^2 + 4^2)) = 0.6, θ = 53.13 degrees.

If vector A = 2i + 3j and vector B = 3i + 4j, what is the angle between them? (2023)

If vector A = 2i + 3j and vector B = 3i + 4j, what is the angle between them? (2023)

Step 1: Identify the components of vector A and vector B. Vector A = 2i + 3j and vector B = 3i + 4j.
Step 2: Calculate the dot product of vectors A and B. The dot product A · B = (2 * 3) + (3 * 4).
Step 3: Simplify the dot product calculation. A · B = 6 + 12 = 18.
Step 4: Calculate the magnitude of vector A. |A| = √(2^2 + 3^2) = √(4 + 9) = √13.
Step 5: Calculate the magnitude of vector B. |B| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
Step 6: Use the formula for the cosine of the angle θ between the vectors: cos(θ) = (A · B) / (|A| * |B|).
Step 7: Substitute the values into the formula: cos(θ) = 18 / (√13 * 5).
Step 8: Calculate the denominator: √13 * 5 ≈ 14.42.
Step 9: Calculate cos(θ): cos(θ) ≈ 18 / 14.42 ≈ 0.6.
Step 10: Find the angle θ by taking the inverse cosine: θ = cos⁻¹(0.6).
Step 11: Calculate θ using a calculator: θ ≈ 53.13 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 each vector is essential for determining the angle.
Trigonometric Functions – Understanding how to use the cosine function to find angles from the dot product.