 # Equation Of A Line In Three Dimensions

Equation of a line is defined as y= mx+c, where c is the y-intercept and m is the slope. Vectors can be defined as a quantity possessing both direction and magnitude. Position vectors simply denote the position or location of a point in the three-dimensional Cartesian system with respect to a reference origin. Further we shall study in detail about vectors and Cartesian equation of a line in three dimensions. It is known that we can uniquely determine a line if:

• It passes through a particular point in a specific direction, or
• It passes through two unique points

Let us study each case separately and try to determine the equation of a line in both the given cases.

## Equation of a Line passing through a point and parallel to a vector

Let us consider that the position vector of the given point be  $\vec{a}$with respect to the origin. The line passing through point A is given by l and it is parallel to the vector $\vec{k}$ as shown below. Let us choose any random point R on the line l and its position vector with respect to origin of the rectangular co-ordinate system is given by $\vec{r}$. Since the line segment, $\overline{AR}$ is parallel to vector  $\vec{k}$, therefore for any real number α,

$\overline{AR}$ = α $\vec{k}$

Also, $\overline{AR}$=$\overline{OR}$ – $\overline{OA}$

Therefore, α $\vec{r}$ = $\vec{r}$$\vec{a}$

From the above equation it can be seen that for different values of α, the above equations give the position of any arbitrary point R lying on the line passing through point A and parallel to vector k. Therefore, the vector equation of a line passing through a given point and parallel to a given vector is given by:

$\vec{r}$ = $\vec{a}$ + α$\vec{k}$

If the three dimensional coordinates of the point A are given as (x1, y1, z1) and the direction cosines of this point is given as a, b, c then considering the rectangular coordinates of point R as (x, y, z) Substituting these values in the vector equation of a line passing through a given point and parallel to a given vector and equating the coefficients of unit vectors i, j and k, we have,

< Eliminating α we have: This gives us the Cartesian equation of line.

Equation of a Line passing through two given points

Let us consider that the position vector of the two given points A and B be $\vec{a}$ and $\vec{b}$ with respect to the origin. Let us choose any random point R on the line and its position vector with respect to origin of the rectangular co-ordinate system is given by $\vec{r}$ . Point R lies on the line AB if and only if the vectors  $\overline{AR}$ and $\overline{AB}$ are collinear. Also,

$\overline{AR}$$\vec{r}$$\vec{a}$

$\overline{AB}$$\vec{b}$$\vec{a}$

Thus R lies on AB only if;

$\vec{r}$$\vec{a}$ = α $\vec{b}$$\vec{a}$

Here α is any real number.

From the above equation it can be seen that for different values of α, the above equation gives the position of any arbitrary point R lying on the line passing through point A and B. Therefore, the vector equation of a line passing through two given points is given by:

$\vec{r}$ = $\vec{a} + α ($\vec{b} \) – $\vec{a}$)

If the three dimensional coordinates of the points A and B are given as (x1, y1, z1) and (x2, y2, z2) then considering the rectangular coordinates of point R as (x, y, z) Substituting these values in the vector equation of a line passing through two given points and equating the coefficients of unit vectors i, j and k, we have Eliminating α we have: This gives us the Cartesian equation of a line.