Parametric equation of a line

Question

When we want to find the line that goes through two points $(-2, -2, -2)$ and $(2, -2, 4)$ we use the formula $$\overrightarrow{l}(t)=\overrightarrow{a}+t(\overrightarrow{b}-\overrightarrow{a}), t \in \mathbb{R}$$ right??

Do we get different lines when we use as $\overrightarrow{a}$ the point $(-2, -2, -2)$ and when we use as $\overrightarrow{a}$ the point $(2, -2, 4)$ ??

Also why is the formula $\overrightarrow{l}(t)=\overrightarrow{a}+t(\overrightarrow{b}-\overrightarrow{a})$ and not $\overrightarrow{l}(t)=\overrightarrow{a}+t(\overrightarrow{a}-\overrightarrow{b})$ ??

Answer 1

No, you won't get a different line. In fact you should read (or remember) the standard equation for a line as: "an arbitrary point on the line (so you can choose to use either $\vec a$ or $\vec b$ - or any other known point! - here) plus multiples (that's the parameter $t$) of any direction vector".

If a direction vector isn't given, you can always 'make' one by subtracting any two different points on the line, so you can choose to take either $\vec b - \vec a$ or $\vec a - \vec b$. Any non-zero multiple of a direction vector is again a suitable direction vector: $\vec b - \vec a$ and $\vec a - \vec b$ are just opposite vectors.

In other words, if $\vec a$ and $\vec b$ represent two different points in space, the line going through both could be represented as any of the following:

• $\vec l (t) = \vec a + t (\vec b - \vec a)$
• $\vec l (t) = \vec a + t (\vec a - \vec b)$
• $\vec l (t) = \vec b + t (\vec b - \vec a)$
• $\vec l (t) = \vec b + t (\vec a - \vec b)$
• $\cdots$

Answer 2

We get same direction in space but different, opposite sense.

In the formula $\overrightarrow{l}(t)=\overrightarrow{a}+t(\overrightarrow{b}-\overrightarrow{a})$

for t=o

$ \overrightarrow{l}(t)=\overrightarrow{a} $ at the base of arrow.

and for t=1

$ \overrightarrow{l}(t)=\overrightarrow{b} $ at the tip of arrow.