# What is the vector equation of a straight line?

I know that, the vector equation of a line passing through $(x_0, y_0, z_0)$ is,

$\vec r = \vec r _0 + t \vec v$

Where, $\vec r$ = the vector for the subject line.

$\vec r_0$ = a position vector that points to the direction of the point $(x_0, y_0, z_0)$.

$\vec v$ = a vector which is parallel to our subject straight line.

But, I have two questions here:

(1) What is $t$ and how can we find the value of $t$?

(2) What is the origin of $\vec r_0$?

(3) If $r = r_0 + tv$ Is the parametric equation of a line, then what is the difference between a parametric equation and a vector equation?

• $t$ is a parameter; it may be any real number. Origin of $\vec r_0$ is $(0,0,0)$, of course Commented Jul 25, 2015 at 17:23
• for $t$ we have $-\infty<t<\infty$ Commented Jul 25, 2015 at 17:24

The equation $$\vec r = \vec r_0 + t \vec v$$ is a parametric equation for a line. You can see $\vec r$ as a function of $t$: $$\vec r(t) = r_0 + tv$$ and for any specific $t_0$, $\vec r(t_0)$ will be a vector pointing to a point on the line.

The whole line is the set of all points which are produced by $\vec r(t)$, i.e. the set $$\{\vec r(t) \mid t \in \mathbb R\}.$$

Thus, it does not make sense to ask "how to solve for $t$?" as your question is phrased.

However, if you have a point $\vec p$ and ask the question "for which $t$ is $\vec r(t) = \vec p$?", that does make sense. In this case you solve the equation $$\vec p = \vec r(t) = \vec r_0 + t \vec v$$ and this will be solvable only when $\vec p$ is on the line.

• I am lost in the middle of vectors and scalars. You haven't use $\vec r$ sign for the symbols $r, r_0$ and $v$. Why? Aren't they vectors? if $r = r_0 + tv$ Is the parametric equation of a line, then what is the difference between a parametric equation and a vector equation?
– user6704
Commented Jul 25, 2015 at 17:49
• @BROY, I have added some vector notation. I didn't use them because it was obvious from the context which was which (I usually don't use $\vec {}$ because it is is clunky, furthermore you didn't use them in your original post). I am not sure what you mean by a vector equation, could you elaborate? Commented Jul 25, 2015 at 18:05
• " furthermore you didn't use them in your original post" ---- I was in the process of editing them. It took time, coz, I am not proficient with MathJax. I think, you posted answer, when I was editing them.
– user6704
Commented Jul 25, 2015 at 18:13
• ,,, Are parametric and vector equations same or different?
– user6704
Commented Jul 25, 2015 at 18:15
• They are different. The link in the answer explains what a parametric equation is. Commented Jul 25, 2015 at 18:19