For three points α, β, and γ:
・The three points are collinear ⇔ (γ−α)/(β−α) is a real number
・αβ ⊥ αγ ⇔ (γ−α)/(β−α) is a pure imaginary number
Example ① Calculate (γ−α)/(β−α) when α=0, β=1, and γ=i.
Solution:i/1 = i. Since it is pure imaginary, αβ ⊥ αγ; since |γ|/|β|=1, the triangle is a right isosceles triangle.
Solution:i/1 = i. Since it is pure imaginary, αβ ⊥ αγ; since |γ|/|β|=1, the triangle is a right isosceles triangle.
Key Point
- The angle and ratio of lengths can be determined from the argument and absolute value of (γ−α)/(β−α)
Practice Problems
- Are α=1, β=3+2i, and γ=−1+4i collinear?
Answers and Explanations
- Answer:They are not collinear
(γ−α)/(β−α)=(−2+4i)/(2+2i)=(1+3i)/2, which is not real.