The angle between the vectors A = i + j and B = j + k is:

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Question: The angle between the vectors A = i + j and B = j + k is:

Options:

Correct Answer: 90°

Solution:

cos(θ) = (A · B) / (|A||B|) = (0) / (√2 * √2) = 0, thus θ = 90°.

The angle between the vectors A = i + j and B = j + k is:

The angle between the vectors A = i + j and B = j + k is:

Step 1: Identify the vectors A and B. A = i + j and B = j + k.
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, 0) and B = (0, 1, 1), the dot product A · B = (1*0) + (1*1) + (0*1) = 0 + 1 + 0 = 1.
Step 4: Calculate the magnitudes of A and B. The magnitude of A = |A| = √(1^2 + 1^2 + 0^2) = √2. The magnitude of B = |B| = √(0^2 + 1^2 + 1^2) = √2.
Step 5: Use the formula for the cosine of the angle θ: cos(θ) = (A · B) / (|A||B|). Substitute the values: cos(θ) = 1 / (√2 * √2) = 1 / 2.
Step 6: Find the angle θ by taking the inverse cosine: θ = cos⁻¹(0) = 90°.

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.