Find the angle θ between vectors a = i + j - k and b = i - j + k.
Answer and explanation
Correct answer: D. cos^(-1)(-1/3)
- Step 1: a · b = 1(1) + 1(-1) + (-1)(1) = 1 - 1 - 1 = -1.
- Step 2: |a| = sqrt(1^2 + 1^2 + (-1)^2) = √3.
- Step 3: |b| = sqrt(1^2 + (-1)^2 + 1^2) = √3.
- Step 4: cos θ = (a · b) / (|a| |b|) = -1 / (√3 * √3) = -1/3 => θ = cos^(-1)(-1/3).
Why not the others: The other values (cos^(-1)(1/3), cos^(-1)(-2/3), cos^(-1)(2/3)) arise from summing vector components linearly without squaring, or forgetting to take the square root of the sum.
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