If A = 3i + 4j and B = 4i + 3j, find the angle between A and B.

If A = 3i + 4j and B = 4i + 3j, find the angle between A and B.

Step 1: Identify the vectors A and B. A = 3i + 4j and B = 4i + 3j.
Step 2: Calculate the dot product A · B. This is done by multiplying the corresponding components: (3 * 4) + (4 * 3).
Step 3: Perform the multiplication: 3 * 4 = 12 and 4 * 3 = 12. Now add them together: 12 + 12 = 24.
Step 4: Calculate the magnitude of vector A, |A|. Use the formula |A| = √(3^2 + 4^2).
Step 5: Calculate 3^2 = 9 and 4^2 = 16. Now add them: 9 + 16 = 25. Take the square root: √25 = 5.
Step 6: Calculate the magnitude of vector B, |B|. Use the formula |B| = √(4^2 + 3^2).
Step 7: Calculate 4^2 = 16 and 3^2 = 9. Now add them: 16 + 9 = 25. Take the square root: √25 = 5.
Step 8: Use the dot product and magnitudes to find cos(θ). The formula is cos(θ) = (A · B) / (|A| * |B|).
Step 9: Substitute the values: cos(θ) = 24 / (5 * 5) = 24 / 25.
Step 10: To find the angle θ, use the inverse cosine function: θ = cos^(-1)(24/25).
Step 11: Calculate θ using a calculator: θ ≈ 60°.

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 finding the angle between them.
Inverse Cosine Function – Using the inverse cosine function to determine the angle from the cosine value.