Determine the angle theta between vectors a = <1, 1> and b = <0, 2>.
Answer and explanation
Correct answer: B. 45 degrees
- Step 1: Calculate dot product: a . b = 1(0) + 1(2) = 2.
- Step 2: Calculate magnitude ||a|| = sqrt(1^2 + 1^2) = sqrt(2).
- Step 3: Calculate magnitude ||b|| = sqrt(0^2 + 2^2) = 2.
- Step 4: Use cosine formula: cos(theta) = (a . b) / (||a|| ||b||) = 2 / (2 sqrt(2)) = 1 / sqrt(2).
- Step 5: Find angle: theta = arccos(1 / sqrt(2)) = 45 degrees.
Answer
45 degrees.
Why not the others: '90 degrees' assumes perpendicularity when dot product is non-zero; '60 degrees' corresponds to cos(theta) = 1/2; '30 degrees' corresponds to cos(theta) = sqrt(3)/2.
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