If A = 2i + 2j and B = 3i + 3j, what is the angle between A and B?

If A = 2i + 2j and B = 3i + 3j, what is the angle between A and B?

Step 1: Identify the vectors A and B. A = 2i + 2j and B = 3i + 3j.
Step 2: Calculate the dot product A · B. This is done by multiplying the corresponding components: (2 * 3) + (2 * 3) = 6 + 6 = 12.
Step 3: Calculate the magnitude of vector A. |A| = √(2^2 + 2^2) = √(4 + 4) = √8 = 2√2.
Step 4: Calculate the magnitude of vector B. |B| = √(3^2 + 3^2) = √(9 + 9) = √18 = 3√2.
Step 5: Use the formula for the cosine of the angle θ between two vectors: cos(θ) = (A · B) / (|A| * |B|). Substitute the values: cos(θ) = 12 / (2√2 * 3√2).
Step 6: Simplify the denominator: 2√2 * 3√2 = 6 * 2 = 12. So, cos(θ) = 12 / 12 = 1.
Step 7: Find the angle θ. If cos(θ) = 1, then θ = 0°.

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 determining the angle between them.
Cosine of Angle – Understanding the relationship between the dot product and the cosine of the angle is crucial for solving the problem.