17. Parametric Line

Set 𝐿 of all points (i.e., line) equal to the set of all vectors 𝑥⃗ plus some scalar 𝑡 times the vector 𝑣⃗ such that 𝑡 can be any real number (ℝ)

𝐿={𝑥⃗+𝑡𝑣⃗|𝑡∈ℝ}
Example
𝑎⃗=[21]𝑏⃗=[03]𝑡=1

The line 𝐿 can be defined as:

𝑏⃗+𝑡(𝑏⃗−𝑎⃗)=[03]+1([03]−[21])=[03]+1[−22]=[−25]

The line 𝐿 can also be defined as:

𝑎⃗+𝑡(𝑏⃗−𝑎⃗)=[21]+1([03]−[21])=[21]+1[−22]=[03]

Generalization

𝑃1=[𝑥1𝑥2⋮𝑥𝑛]𝑃2=[𝑥1𝑥2⋮𝑥𝑛]𝐿=𝑃1+𝑡(𝑃1−𝑃2)|𝑡∈ℝ
Example
𝑃1⃗=[−12]𝑃2⃗=[03]𝑃1⃗−𝑃2⃗=[−1−1]
  • 𝐿 starts at 𝑃1 and moves toward 𝑃2 as 𝑡 decreases
𝐿=𝑃1+𝑡(𝑃1−𝑃2)=[−12]+𝑡[−1−1]𝑥=−1+𝑡(−1)𝑦=2+𝑡(−1)
  • 𝐿 starts at 𝑃2 and moves toward 𝑃1 as 𝑡 increases
𝐿=𝑃2+𝑡(𝑃1−𝑃2)=[03]+𝑡[−1−1]𝑥=0+𝑡(−1)𝑦=3+𝑡(−1)
  • 𝐿 starts at 𝑃1 and moves toward 𝑃2 as 𝑡 increases
𝐿=𝑃1+𝑡(𝑃2−𝑃1)=[−12]+𝑡[11]𝑥=−1+𝑡(1)𝑦=2+𝑡(1)
  • 𝐿 starts at 𝑃2 and moves toward 𝑃1 as 𝑡 decreases
𝐿=𝑃2+𝑡(𝑃2−𝑃1)=[03]+𝑡[11]𝑥=0+𝑡(1)𝑦=3+𝑡(1)