Cartesian Equation of a Line

Here you will learn cartesian equation of line in 3d passing through a fixed point and passing through two points.

Let’s begin –

Cartesian Equation of a Line

The cartesian equations of a straight line passing through a fixed point $$(x_1, y_1, z_1)$$ having direction ratios proportional to a, b, c is given by

$$x – x_1\over a$$ = $$y – y_1\over b$$ = $$z – z_1\over c$$

Remark 1 : The above form of a line is known as the symmetrical form of a line.

Remark 2 : The parametric equations of the line $$x – x_1\over a$$ = $$y – y_1\over b$$ = $$z – z_1\over c$$ are

x = $$x_1 + a\lambda$$, y = $$y_1 + b\lambda$$, z = $$z_1 + c\lambda$$, where $$\lambda$$ is the parameter.

Remark 3 : The coordinates of any point on the line $$x – x_1\over a$$ = $$y – y_1\over b$$ = $$z – z_1\over c$$ are

($$x_1 + a\lambda$$,  $$y_1 + b\lambda$$, $$z_1 + c\lambda$$), where $$\lambda$$ $$\in$$ R.

Remark 4 : Since the direction cosines of a line are also its direcion ratios. Therefore, equations of a line passing through $$(x_1, y_1, z_1)$$ and having direction cosines l, m, n are

$$x – x_1\over l$$ = $$y – y_1\over m$$ = $$z – z_1\over n$$

Remark 5 : Since x, y and z-axes passes through the origin and have direction cosines 1, 0, 0; 0, 1, 0 and 0, 0, 1 respectively. Therefore, their equations are

x-axis : $$x – 0\over 1$$ = $$y – 0\over 0$$ = $$z – 0\over 0$$ or, y = 0 and z = 0

y-axis : $$x – 0\over 0$$ = $$y – 0\over 1$$ = $$z – 0\over 0$$ or, x = 0 and z = 0

z-axis : $$x – 0\over 0$$ = $$y – 0\over 0$$ = $$z – 0\over 1$$ or, y = 0 and y = 0

Cartesian Equation of a Line Passing Through Two Points

The Cartesian equation of aline passing through two given points $$(x_1, y_1, z_1)$$ and $$(x_2, y_2, z_2)$$ is given by

$$x – x_1\over x_2 – x_1$$ = $$y – y_1\over y_2 – y_1$$ = $$z – z_1\over z – z_1$$

Example : Find the cartesian equation of line passing through A(3, 4, -7) and B(1, -1, 6).

Solution : We have, A(3, 4, -7) and B(1, -1, 6)

The cartesian equation of line passing through two points is

$$x – x_1\over x_2 – x_1$$ = $$y – y_1\over y_2 – y_1$$ = $$z – z_1\over z – z_1$$

= $$x – 3\over -2$$ = $$y – 4\over -5$$ = $$z + 7\over 13$$