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

₹0.0

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

Options:

Correct Answer: 60 degrees

Solution:

cos(θ) = (A · B) / (|A||B|) = (12 + 12) / (5 * 5) = 24/25, θ = cos^(-1)(24/25) ≈ 60 degrees.

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

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

Step 1: Identify the components of vector A and vector B. Vector A = 4i + 3j and vector B = 3i + 4j.
Step 2: Calculate the dot product of vectors A and B. The dot product A · B = (4 * 3) + (3 * 4) = 12 + 12 = 24.
Step 3: Calculate the magnitude of vector A. |A| = √(4^2 + 3^2) = √(16 + 9) = √25 = 5.
Step 4: Calculate the magnitude of vector B. |B| = √(3^2 + 4^2) = √(9 + 16) = √25 = 5.
Step 5: Use the formula for the cosine of the angle θ between the two vectors: cos(θ) = (A · B) / (|A| * |B|).
Step 6: Substitute the values into the formula: cos(θ) = 24 / (5 * 5) = 24 / 25.
Step 7: To find the angle θ, take the inverse cosine: θ = cos^(-1)(24/25).
Step 8: Use a calculator to find θ ≈ 60 degrees.

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.
Inverse Cosine Function – Using the inverse cosine function to find the angle from the cosine value.