If the angle between vectors A = 2i + 3j and B = 4i + 5j is 60 degrees, find A ·

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Question: If the angle between vectors A = 2i + 3j and B = 4i + 5j is 60 degrees, find A · B.

Options:

Correct Answer: 25

Solution:

A · B = |A||B|cos(60°) = √(2^2 + 3^2) * √(4^2 + 5^2) * 1/2 = √13 * √41 * 1/2 = 20.

If the angle between vectors A = 2i + 3j and B = 4i + 5j is 60 degrees, find A · B.

If the angle between vectors A = 2i + 3j and B = 4i + 5j is 60 degrees, find A · B.

Step 1: Identify the vectors A and B. A = 2i + 3j and B = 4i + 5j.
Step 2: Find the magnitudes of vectors A and B. The magnitude of A is |A| = √(2^2 + 3^2) and the magnitude of B is |B| = √(4^2 + 5^2).
Step 3: Calculate |A|. |A| = √(4 + 9) = √13.
Step 4: Calculate |B|. |B| = √(16 + 25) = √41.
Step 5: Use the formula for the dot product: A · B = |A||B|cos(θ), where θ is the angle between the vectors. Here, θ = 60 degrees.
Step 6: Calculate cos(60°). cos(60°) = 1/2.
Step 7: Substitute the values into the dot product formula: A · B = √13 * √41 * (1/2).
Step 8: Calculate the final result: A · B = (√13 * √41) / 2.
Step 9: Simplify the expression: A · B = 20.

Dot Product of Vectors – The dot product of two vectors can be calculated using the formula A · B = |A||B|cos(θ), where θ is the angle between the vectors.
Magnitude of Vectors – The magnitude of a vector A = ai + bj is calculated as |A| = √(a² + b²).
Cosine of Angles – Understanding the cosine function, particularly cos(60°) = 1/2, is crucial for solving problems involving angles between vectors.