Find the angle between the vectors A = i + j and B = i - j.

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Question: Find the angle between the vectors A = i + j and B = i - j.

Options:

Correct Answer: 90 degrees

Solution:

cos(θ) = (A · B) / (|A||B|) = (1 - 1) / (√2 * √2) = 0, θ = 90 degrees.

Find the angle between the vectors A = i + j and B = i - j.

Find the angle between the vectors A = i + j and B = i - j.

Step 1: Identify the vectors A and B. A = i + j and B = i - j.
Step 2: Calculate the dot product of A and B. A · B = (1)(1) + (1)(-1) = 1 - 1 = 0.
Step 3: Calculate the magnitude of vector A. |A| = √(1^2 + 1^2) = √2.
Step 4: Calculate the magnitude of vector B. |B| = √(1^2 + (-1)^2) = √2.
Step 5: Use the formula for the cosine of the angle θ: cos(θ) = (A · B) / (|A||B).
Step 6: Substitute the values into the formula: cos(θ) = 0 / (√2 * √2) = 0.
Step 7: Find the angle θ using the inverse cosine function: θ = cos⁻¹(0).
Step 8: Determine that 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 finding the angle between them.
Angle Between Vectors – Understanding how to derive the angle from the cosine value is crucial.