Find the angle between the vectors A = i + j and B = j - i. (2022)

Find the angle between the vectors A = i + j and B = j - i. (2022)

Step 1: Identify the vectors A and B. A = i + j and B = j - i.
Step 2: Calculate the dot product A · B. This is done by multiplying the corresponding components of A and B and adding them together.
Step 3: For A = (1, 1) and B = (-1, 1), calculate A · B = (1 * -1) + (1 * 1) = -1 + 1 = 0.
Step 4: Find the magnitudes of A and B. The magnitude of A is |A| = √(1^2 + 1^2) = √2. The magnitude of B is |B| = √((-1)^2 + 1^2) = √2.
Step 5: Use the formula cos(θ) = (A · B) / (|A| |B|). Substitute the values: cos(θ) = 0 / (√2 * √2) = 0.
Step 6: Since cos(θ) = 0, this 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 using the cosine formula.
Orthogonal Vectors – Understanding that a dot product of zero indicates that the vectors are orthogonal (90 degrees apart).