If A = 2i + 3j and B = 4i + 5j, find the angle θ between A and B if A · B = |A||

If A = 2i + 3j and B = 4i + 5j, find the angle θ between A and B if A · B = |A||B|cos(θ).

If A = 2i + 3j and B = 4i + 5j, find the angle θ between A and B if A · B = |A||B|cos(θ).

Step 1: Identify the vectors A and B. A = 2i + 3j and B = 4i + 5j.
Step 2: Calculate the dot product A · B. Use the formula A · B = (2 * 4) + (3 * 5).
Step 3: Perform the multiplication: 2 * 4 = 8 and 3 * 5 = 15.
Step 4: Add the results from Step 3: 8 + 15 = 23. So, A · B = 23.
Step 5: Calculate the magnitude of vector A, |A|. Use the formula |A| = √(2^2 + 3^2).
Step 6: Calculate 2^2 = 4 and 3^2 = 9. Then, add them: 4 + 9 = 13.
Step 7: Find the square root: |A| = √13.
Step 8: Calculate the magnitude of vector B, |B|. Use the formula |B| = √(4^2 + 5^2).
Step 9: Calculate 4^2 = 16 and 5^2 = 25. Then, add them: 16 + 25 = 41.
Step 10: Find the square root: |B| = √41.
Step 11: Use the formula A · B = |A||B|cos(θ) to find cos(θ).
Step 12: Substitute the values: 23 = (√13)(√41)cos(θ).
Step 13: Solve for cos(θ): cos(θ) = 23 / (√13 * √41).

Dot Product – Understanding the dot product of two vectors and its relation to the angle between them.
Magnitude of Vectors – Calculating the magnitude of vectors using the Pythagorean theorem.
Cosine of Angle – Using the cosine function to relate the dot product and magnitudes of vectors to the angle between them.