If vector A = 5i + 5j and vector B = 5i - 5j, what is the angle between A and B?

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Question: If vector A = 5i + 5j and vector B = 5i - 5j, what is the angle between A and B?

Options:

Correct Answer: 90 degrees

Solution:

cos(θ) = (A · B) / (|A||B|) = (25 - 25) / (√(50) * √(50)) = 0, θ = 90 degrees.

If vector A = 5i + 5j and vector B = 5i - 5j, what is the angle between A and B?

If vector A = 5i + 5j and vector B = 5i - 5j, what is the angle between A and B?

Step 1: Identify the vectors A and B. A = 5i + 5j and B = 5i - 5j.
Step 2: Calculate the dot product of A and B. A · B = (5i + 5j) · (5i - 5j).
Step 3: Use the formula for dot product: A · B = (5 * 5) + (5 * -5) = 25 - 25 = 0.
Step 4: Calculate the magnitudes of A and B. |A| = √(5^2 + 5^2) = √(25 + 25) = √50.
Step 5: Calculate the magnitude of B, which is the same as A: |B| = √50.
Step 6: Use the formula for cosine of the angle: cos(θ) = (A · B) / (|A||B) = 0 / (√50 * √50).
Step 7: Since the numerator is 0, cos(θ) = 0, which means θ = 90 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 is essential for determining the angle between them.
Angle Between Vectors – Understanding how to derive the angle from the dot product and magnitudes of vectors.