How to interpret a 3D line equation with zero denominators

[Krushnaraj Pandya]

Homework Statement

I noticed that for a line parallel to z axis the equation is (x-a)/0 = (y-b)/0 = (z-c)/k = t.

Homework Equations

Any 3d geometry equations

The Attempt at a Solution

I can't quite grasp how to visualize this, also I can't see any constant like a 2d line parallel to x-axis would have y=constant. Another problem is how to take a general point on this line with parameter t, do we just say x=a, y=b or ignore the zero...intuitively it seems for a given point x and y have to be a and b to be able to have the line parallel to z. Any good ways to visualize all this?

 

[fresh_42]

Homework Statement

I noticed that for a line parallel to z axis the equation is (x-a)/0 = (y-b)/0 = (z-c)/k = t.

Homework Equations

Any 3d geometry equations

The Attempt at a Solution

I can't quite grasp how to visualize this, also I can't see any constant like a 2d line parallel to x-axis would have y=constant. Another problem is how to take a general point on this line with parameter t, do we just say x=a, y=b or ignore the zero...intuitively it seems for a given point x and y have to be a and b to be able to have the line parallel to z. Any good ways to visualize all this?

First of all: a denominator $0$ is nonsense. Just nonsense. Your equation for a line in $\mathbb{R}^3$ is probably: $\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}m_x\\m_y\\m_z\end{bmatrix}\cdot t + \begin{bmatrix}x_0\\y_0\\z_0\end{bmatrix}$ and there is nowhere a division by zero, although any constant there is allowed to be zero as e.g. $m_x=m_y=0$ for a line parallel to the $z-$axis.

You can draw those lines and planes in a coordinate system like

 

[pasmith]

It is common to describe a line in $\mathbb{R}^3$ for which the tangent vector has no zero component by taking the components of the vector equation and setting the three resulting expressions for $t$ equal to each other: $$\frac{x - x_0}{m_x} = \frac{y - y_0}{m_y} = \frac{z - z_0}{m_z}.$$ But if any of $m_x$, $m_y$ and $m_z$ are zero then you can't do that, and you have to write separate equations. For example, if $m_x = 0$ you must write $$x = x_0,\qquad\frac{y - y_0}{m_y} = \frac{z - z_0}{m_z}.$$ If $m_x = m_y = 0$ then you must write $$x = x_0,\qquad y = y_0.$$If $m_x = m_y = m_z = 0$ then you have a single point, not a line.

 

[Mark44]

I noticed that for a line parallel to z axis the equation is (x-a)/0 = (y-b)/0 = (z-c)/k = t.

First of all: a denominator 0 is nonsense. Just nonsense.

I beg to differ. The equations above are the symmetric-form equations of a line in $\mathbb R^3$. The denominators here should be considered to be notation, but not taken literally to mean division by zero.

From a note in "Calculus and Analytic Geometry, Second Ed.," by Abraham Schwartz, p. 590 (italics added):

Remark 2
If one of the components of $\vec v$ is 0, then one of the members of the symmetric-form equations for a line L in $\vec v's$ direction will have 0 as its denominator. In such a case, the symmetric-form description of L is to be interpreted as a statement about proportional trios of numbers.

I can't quite grasp how to visualize this, also I can't see any constant like a 2d line parallel to x-axis would have y=constant. Another problem is how to take a general point on this line with parameter t, do we just say x=a, y=b or ignore the zero...intuitively it seems for a given point x and y have to be a and b to be able to have the line parallel to z. Any good ways to visualize all this?

In your equations, a vector $\vec v$ in the direction of the line is <0, 0, k>; that is, it is a vector parallel to the z-axis.

 

[fresh_42]

I beg to differ. The equations above are the symmetric-form equations of a line in $\mathbb R^3$. The denominators here should be considered to be notation, but not taken literally to mean division by zero.

Quite an unfortunate way to express it, there are better representations.

 

[Mark44]

Quite an unfortunate way to express it, there are better representations.

That may be, but the symmetric-form equations are a compact way to incorporate a point on the line and its direction.

 

[SammyS]

Quite an unfortunate way to express it, there are better representations.

I agree, except that the word "unfortunate" is much too mild.