If vector A = 2i + 3j and vector B = 4i + 5j, what is the angle between A and B?

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Question: If vector A = 2i + 3j and vector B = 4i + 5j, what is the angle between A and B? (2023)

Options:

Correct Answer: 60 degrees

Exam Year: 2023

Solution:

cos(θ) = (A · B) / (|A||B|) = (8 + 15) / (√(13) * √(41)). θ = 60 degrees.

If vector A = 2i + 3j and vector B = 4i + 5j, what is the angle between A and B? (2023)

If vector A = 2i + 3j and vector B = 4i + 5j, what is the angle between A and B? (2023)

Step 1: Identify the components of vector A and vector B. Vector A = 2i + 3j and vector B = 4i + 5j.
Step 2: Calculate the dot product of vectors A and B. The dot product A · B = (2 * 4) + (3 * 5) = 8 + 15 = 23.
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| = √(4^2 + 5^2) = √(16 + 25) = √41.
Step 5: Use the formula for the cosine of the angle θ between the two vectors: cos(θ) = (A · B) / (|A||B|).
Step 6: Substitute the values into the formula: cos(θ) = 23 / (√13 * √41).
Step 7: Calculate the angle θ using the inverse cosine function: θ = cos⁻¹(23 / (√13 * √41)).
Step 8: Find the angle θ, which is approximately 60 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 between them.
Trigonometric Functions – Understanding how to use cosine to find angles from the dot product.