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

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

Options:

Correct Answer: 90 degrees

Solution:

cos(θ) = (A · B) / (|A||B|) = (25 - 25) / (5√2 * 5√2) = 0, θ = 90 degrees.

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

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

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 = (5)(5) + (5)(-5) = 25 - 25 = 0.
Step 3: Calculate the magnitude of vector A. |A| = √(5^2 + 5^2) = √(25 + 25) = √50 = 5√2.
Step 4: Calculate the magnitude of vector B. |B| = √(5^2 + (-5)^2) = √(25 + 25) = √50 = 5√2.
Step 5: Use the formula for the cosine of the angle θ: cos(θ) = (A · B) / (|A||B).
Step 6: Substitute the values: cos(θ) = 0 / (5√2 * 5√2) = 0.
Step 7: Find the angle θ using the inverse cosine: θ = cos⁻¹(0) = 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.