Question

Prove that every line in R^3 is a regular curve.

Answer 1

Every line in
`\mathbb{R}^(3)` is a function of the form
`\gamma (t)={\bf p}+t {\bf v}`, where
`{\bf p}` is point where the line passes and
`{\bf v}` is a nonzero vector which is called the direction vector of the line. Then, if we derive the function
`\gamma` we obtain
`\gamma'(t)={\bf v} \\eq (0,0,0)`, so
`\gamma(t)={\bf p}+t {\bf v}` is a regular curve.

Explanation:

Every line in
`\mathbb{R}^(3)` can be parametrized by

`\gamma (t)={\bf p}+t{\bf v}=(p_(1),p_(2),p_(3))+t(v_(1),v_(2),v_(3))=(p_(1)+tv_(1),p_(2)+tv_(2),p_(3)+tp_(3))`, where
`t\in \mathbb{R}`. To derivate the function
`\gamma` we only need to derive each component. Then we have that

`\gamma'(t)=((d)/(dt)(p_(1)+tv_(1)),(d)/(dt)(p_(2)+tv_(2)),(d)/(dt)(p_(3)+tv_(3)))=(v_(1),v_(2),v_(3))={\bf v}\\eq (0,0,0).`

Now, remember that a a parametrized curve is said to be regular if
`\gamma'\\eq 0` for all
`t`.