Dot product and right angles
a · b = 0 precisely when the vectors are perpendicular (and nonzero).
Key formulas
Computation
(2,−1)·(3,6)=6−6=0, so the vectors are perpendicular.
Angle
If the dot product is not zero, cos θ = (a·b)/(|a||b|) gives the angle between them.
Geometry use
To prove a triangle is right-angled at B, show BA · BC = 0.
Worked example
Worked example
Problem. Are vectors (2, −1) and (3, 6) perpendicular?
Solution. Dot product = 2·3 + (−1)·6 = 6 − 6 = 0. Yes — they are perpendicular.
Sign
Acute angles give positive dot products; obtuse give negative.
Review quiz
Are (5,4) and (−4,5) perpendicular?
Practice questions
- Are (4,2) and (−1,2) perpendicular?
- Find cos θ for (1,0) and (1,1).
Study notes
In 3D?
Same rule: include the third components in the sum.