Angles between vectors
Objective. Use dot products to determine angles and perpendicularity.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
The angle between two vectors a and b is found using the dot product rule cos θ = (a·b)/(|a||b|), where a·b is the sum of corresponding component products and |a|, |b| are magnitudes. Two vectors are perpendicular exactly when a·b = 0. For a = (1,2,2) and b = (2,−1,2), a·b = 2−2+4 = 4, |a| = 3, |b| = 3, giving cos θ = 4/9. Always divide by the product of magnitudes, not their sum, and check perpendicularity by testing whether the dot product equals zero.
Worked example
Find the angle between the vectors a = (1,0,1) and b = (0,1,1), to one decimal place.
- 1Compute the dot product: a·b = 1(0)+0(1)+1(1) = 1
- 2Find the magnitudes: |a| = √2, |b| = √2
- 3Apply cos θ = 1/(√2×√2) = 1/2
- 4Find θ = arccos(1/2) = 60.0°
Answer: θ = 60.0°
Concept mastery
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Practice
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Start practisingCommon mistake. Students sometimes compute the dot product correctly but divide by |a|+|b| instead of |a|×|b|, giving an incorrect cosine value. Always multiply the two magnitudes together in the denominator.
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