If A = 5i + 2j and B = 3i + 6j, find the angle θ between A and B if A · B = |A||

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Question: If A = 5i + 2j and B = 3i + 6j, find the angle θ between A and B if A · B = |A||B|cos(θ).

Options:

Correct Answer: 45°

Solution:

A · B = 5*3 + 2*6 = 15 + 12 = 27. |A| = √(5^2 + 2^2) = √29, |B| = √(3^2 + 6^2) = √45. cos(θ) = 27/(√29 * √45). θ = 45°.

If A = 5i + 2j and B = 3i + 6j, find the angle θ between A and B if A · B = |A||B|cos(θ).

If A = 5i + 2j and B = 3i + 6j, find the angle θ between A and B if A · B = |A||B|cos(θ).

Step 1: Identify the vectors A and B. A = 5i + 2j and B = 3i + 6j.
Step 2: Calculate the dot product A · B. Use the formula A · B = (5 * 3) + (2 * 6).
Step 3: Perform the multiplication: 5 * 3 = 15 and 2 * 6 = 12.
Step 4: Add the results from Step 3: 15 + 12 = 27. So, A · B = 27.
Step 5: Calculate the magnitude of vector A, |A|. Use the formula |A| = √(5^2 + 2^2).
Step 6: Calculate 5^2 = 25 and 2^2 = 4. Then, add them: 25 + 4 = 29.
Step 7: Find the square root: |A| = √29.
Step 8: Calculate the magnitude of vector B, |B|. Use the formula |B| = √(3^2 + 6^2).
Step 9: Calculate 3^2 = 9 and 6^2 = 36. Then, add them: 9 + 36 = 45.
Step 10: Find the square root: |B| = √45.
Step 11: Use the formula A · B = |A||B|cos(θ) to find cos(θ). Rearrange to get cos(θ) = A · B / (|A| * |B|).
Step 12: Substitute the values: cos(θ) = 27 / (√29 * √45).
Step 13: Calculate the value of cos(θ).
Step 14: Use the inverse cosine function to find θ: θ = cos⁻¹(cos(θ)).
Step 15: Determine the angle θ, which is approximately 45°.

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.