If vector A = 2i + 3j and vector B = 4i + k, what is the angle between A and B?

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Question: If vector A = 2i + 3j and vector B = 4i + k, what is the angle between A and B?

Options:

Correct Answer: 60 degrees

Solution:

cos(θ) = (A · B) / (|A| |B|). A · B = 8 + 0 + 3(0) = 8; |A| = √(2^2 + 3^2) = √13; |B| = √(4^2 + 1^2) = √17. θ = cos^(-1)(8/(√13 * √17)).

If vector A = 2i + 3j and vector B = 4i + k, what is the angle between A and B?

If vector A = 2i + 3j and vector B = 4i + k, what is the angle between A and B?

Step 1: Identify the components of vector A and vector B. A = 2i + 3j means A has components (2, 3) and B = 4i + k means B has components (4, 0, 1).
Step 2: Calculate the dot product A · B. This is done by multiplying the corresponding components of A and B and adding them together: A · B = (2 * 4) + (3 * 0) + (0 * 1) = 8 + 0 + 0 = 8.
Step 3: Calculate the magnitude of vector A. The magnitude |A| is found using the formula |A| = √(x^2 + y^2) where x and y are the components of A. So, |A| = √(2^2 + 3^2) = √(4 + 9) = √13.
Step 4: Calculate the magnitude of vector B. The magnitude |B| is found using the formula |B| = √(x^2 + y^2 + z^2) where x, y, and z are the components of B. So, |B| = √(4^2 + 0^2 + 1^2) = √(16 + 0 + 1) = √17.
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) = 8 / (√13 * √17).
Step 6: To find the angle θ, take the inverse cosine: θ = cos^(-1)(8 / (√13 * √17)).

Dot Product – Understanding how to calculate the dot product of two vectors.
Magnitude of Vectors – Calculating the magnitude of vectors using the Pythagorean theorem.
Angle Between Vectors – Using the cosine formula to find the angle between two vectors.