Find the angle θ between the vectors A = i + 2j and B = 2i + 3j if A · B = |A||B

Find the angle θ between the vectors A = i + 2j and B = 2i + 3j if A · B = |A||B|cos(θ).

Find the angle θ between the vectors A = i + 2j and B = 2i + 3j if A · B = |A||B|cos(θ).

Step 1: Identify the vectors A and B. A = i + 2j and B = 2i + 3j.
Step 2: Calculate the dot product A · B. This is done by multiplying the corresponding components of A and B: (1 * 2) + (2 * 3) = 2 + 6 = 8.
Step 3: Calculate the magnitude of vector A, |A|. Use the formula |A| = √(x^2 + y^2) where x and y are the components of A. Here, |A| = √(1^2 + 2^2) = √(1 + 4) = √5.
Step 4: Calculate the magnitude of vector B, |B|. Use the same formula: |B| = √(2^2 + 3^2) = √(4 + 9) = √13.
Step 5: Use the formula A · B = |A||B|cos(θ) to find cos(θ). We already found A · B = 8, |A| = √5, and |B| = √13.
Step 6: Substitute the values into the formula: 8 = (√5)(√13)cos(θ).
Step 7: Solve for cos(θ): cos(θ) = 8 / (√5 * √13).

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 find the angle.