If A = 6i + 8j and B = 3i + 4j, what is the angle between A and B?

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Question: If A = 6i + 8j and B = 3i + 4j, what is the angle between A and B?

Options:

Correct Answer: 90°

Solution:

A · B = 6*3 + 8*4 = 18 + 32 = 50. |A| = 10, |B| = 5. cos(θ) = 50/(10*5) = 1. θ = 0°.

If A = 6i + 8j and B = 3i + 4j, what is the angle between A and B?

If A = 6i + 8j and B = 3i + 4j, what is the angle between A and B?

Step 1: Identify the vectors A and B. A = 6i + 8j and B = 3i + 4j.
Step 2: Calculate the dot product A · B. This is done by multiplying the corresponding components of A and B: (6 * 3) + (8 * 4).
Step 3: Perform the multiplication: 6 * 3 = 18 and 8 * 4 = 32.
Step 4: Add the results of the multiplication: 18 + 32 = 50. So, A · B = 50.
Step 5: Calculate the magnitude of vector A, |A|. Use the formula |A| = √(6^2 + 8^2).
Step 6: Calculate 6^2 = 36 and 8^2 = 64. Then, add them: 36 + 64 = 100.
Step 7: Take the square root of 100: |A| = √100 = 10.
Step 8: Calculate the magnitude of vector B, |B|. Use the formula |B| = √(3^2 + 4^2).
Step 9: Calculate 3^2 = 9 and 4^2 = 16. Then, add them: 9 + 16 = 25.
Step 10: Take the square root of 25: |B| = √25 = 5.
Step 11: Use the dot product and magnitudes to find cos(θ): cos(θ) = (A · B) / (|A| * |B|).
Step 12: Substitute the values: cos(θ) = 50 / (10 * 5).
Step 13: Calculate 10 * 5 = 50. So, cos(θ) = 50 / 50 = 1.
Step 14: Find the angle θ using the inverse cosine function: θ = cos⁻¹(1).
Step 15: The angle θ = 0°.

Dot Product – The dot product of two vectors is calculated by multiplying their corresponding components and summing the results.
Magnitude of Vectors – The magnitude of a vector is calculated using the formula |A| = √(x^2 + y^2) for a vector A = xi + yj.
Angle Between Vectors – The angle between two vectors can be found using the formula cos(θ) = (A · B) / (|A| |B|).