Section 1.5 Equations of Lines

1.5 Equations of Lines

In this section, we introduce equations of lines in a 3-dimensional space. Recall that in a 2-dimensional space, given two distinct points

[latex]P=(x_{1},y_{1})[/latex] and [latex]Q=(x_{2},y_{2})[/latex], we can define a line passing those two points by finding the slope of the two points, [latex]\frac{y_{2}-y_{1}}{x_{2}-x_{1}}=m[/latex] and the point-slope formula gives [latex](y-y_{1})=\frac{y_{2}-y_{1}}{x_{2}-x_{1}}(x-x_{1})[/latex]. We rewrite the point-slope formula if [latex]x_{1}\neq x_{2}[/latex] and [latex]y_{1}\neq y_{2}[/latex],

\[\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{x-x_{1}}{x_{2}-x_{1}}=t\]

where [latex]t[/latex] is a new variable which is called a parameter. Then we have [latex]y-y_{1}=t(y_{2}-y_{1})[/latex] and [latex]y=y_{1}+t(y_{2}-y_{1})[/latex]. Similarly, we obtain [latex]x=x_{1}+t(x_{2}-x_{1})[/latex]. Hence any point, [latex](x,y)[/latex], on the line can be presented as [latex](x,y)=(x_{1}+t(x_{2}-x_{1}),y_{1}+t(y_{2}-y_{1})[/latex]. Notice that [latex]\lt x_{2}-x_{1},y_{2}-y_{1}\gt=\overrightarrow{PQ}[/latex] is the vector obtained by the given two points, [latex]P=(x_{1},y_{1})[/latex] and [latex]Q=(x_{2},y_{2})[/latex] and any two points on the same line give a vector and the vector must parallel to the vector [latex]\overrightarrow{PQ}[/latex]. We can use the same construction to find the presentation of points of a line in a 3-dimensional space. 

Definition: 

(a) Given two distinct points in 3-dimensional space, [latex]P=(x_{1},y_{1},z_{1})[/latex] and [latex]Q=(x_{2},y_{2},z_{2})[/latex], one can find a line [latex]L[/latex] passing [latex]P[/latex] and [latex]Q[/latex]. Any point [latex](x,y,z)[/latex] on the line [latex]L[/latex] can be presented as [latex](x,y,z)=(x_{1}+t(x_{2}-x_{1}),y_{1}+t(y_{2}-y_{1}),z_{1}+t(z_{2}-z_{1}))=(x_{1},y_{1},z_{1})+t(x_{2}-x_{1},y_{2}-y_{1},z_{2}-z_{1})[/latex] where [latex]t[/latex] is a parameter. [latex]x=x_{1}+t(x_{2}-x_{1})[/latex], [latex]y=y_{1}+t(y_{2}-y_{1})[/latex] and [latex]z=z_{1}+t(z_{2}-z_{1})[/latex] are called the parameter equations of the line. 

\[\frac{x-x_{1}}{x_{2}-x_{1}}=\frac{y-y_{1}}{y_{2}-y_{1}}=\frac{z-z_{1}}{z_{2}-z_{1}}=t \]

is called the symmetric equations of the line [latex]L[/latex] when [latex]x_{1}\neq x_{2}[/latex], [latex]y_{1}\neq y_{2}[/latex], and [latex]z_{1}\neq z_{2}[/latex].

(b) Given one point in a 3-dimensional space, [latex]P=(x_{1},y_{1},z_{1})[/latex] and a vector [latex]\overrightarrow{v}=(a,b,c)[/latex]. A line [latex]L[/latex] passing [latex]P[/latex] and parallel to [latex]\overrightarrow{v}[/latex] can be presented as [latex](x,y,z)=(x_{1}+at,y_{1}+bt,z_{1}+ct)=(x_{1},y_{1},z_{1})+t(a,b,c)[/latex] where [latex]t[/latex] is a parameter. [latex]x=x_{1}+at[/latex], [latex]y=y_{1}+bt[/latex] and [latex]z=z_{1}+ct[/latex] are called the parameter equations of the line. 

\[\frac{x-x_{1}}{a}=\frac{y-y_{1}}{b}=\frac{z-z_{1}}{c}=t\]

is called the symmetric equations of the line [latex]L[/latex] when neither of [latex]a,[/latex] or [latex]b[/latex] or [latex]c[/latex] are zero.

Example 1: Let [latex]P=(1,-2,3)[/latex] and [latex]Q=(2,1,0)[/latex]. Find the parameter equations and the symmetric equations of the line passing both [latex]P[/latex] and [latex]Q[/latex].

Exercise 1: Let [latex]P=(-1,2,-3)[/latex] and [latex]Q=(0,3,1)[/latex]. Find the parameter equations and the symmetric equations of the line passing both [latex]P[/latex] and [latex]Q[/latex].

Example 2: Let [latex]P=(1,-2,3)[/latex] and [latex]L[/latex] be a line having parameter equations: [latex]x=1+t[/latex], [latex]y=2-t[/latex] and [latex]z=-3+3t[/latex]. Find the parameter equations and the symmetric equations of the line passing [latex]P[/latex] and parallel to the line [latex]L[/latex]. 

Exercise 2: Let [latex]P=(-1,2,0)[/latex] and [latex]L[/latex] be a line having parameter equations: [latex]x=3-t[/latex], [latex]y=2t[/latex] and [latex]z=-1+3t[/latex]. Find the parameter equations and the symmetric equations of the line passing [latex]P[/latex] and parallel to the line [latex]L[/latex]. 

Notice that any point [latex](x,y,z)[/latex] in the space is corresponding to a vector [latex]\overrightarrow{v}=\lt x,y,z\gt[/latex]. The line passing two points [latex](x_{1},y_{1},z_{1})[/latex] and [latex](x_{2},y_{2},z_{2})[/latex], can be written as a vector equation:

\[\lt x,y,z\gt =\lt x_{1}+t(x_{2}-x_{1}),y_{1}+t(y_{2}-y_{1}),z_{1}+t(z_{2}-z_{1})\gt  .\]

Definition:

The line segment from [latex]P=(x_{1},y_{1},z_{1})[/latex] to [latex]Q=(x_{2},y_{2},z_{2})[/latex]

has a vector equation

\[\lt x,y,z\gt=\lt x_{1}+t(x_{2}-x_{1}),y_{1}+t(y_{2}-y_{1}),z_{1}+t(z_{2}-z_{1})\gt \]

where [latex]0 \leq t \leq 1[/latex].

Example 3: Let [latex]P=(1,-2,3)[/latex] and [latex]Q=(2,1,0)[/latex]. Find the vector equation of the line segment from [latex]P[/latex] to [latex]Q[/latex]. How about the vector equation from [latex]Q[/latex] to [latex]P[/latex].

Exercise 3: Let [latex]P=(-1,2,-3)[/latex] and [latex]Q=(0,3,1)[/latex]. Find the vector equation of the line segment from [latex]P[/latex] to [latex]Q[/latex]. How about the vector equation from [latex]Q[/latex] to [latex]P[/latex]?

Example 4: Find the parameter equations and the symmetric equations of the line passing [latex]P=(1,0,-3)[/latex] and is perpendicular to [latex]\overrightarrow{u}=\lt 1,-2,3\gt[/latex] and [latex]\overrightarrow{v}=\lt 2,1,0\gt[/latex].

Exercise 4: Find the parameter equations and the symmetric equations of the line passing [latex]P=(0,-2,1)[/latex] is perpendicular to [latex]\overrightarrow{u}=\lt -1,2,-3\gt[/latex] and [latex]\overrightarrow{v}=\lt -2,0,1\gt[/latex].

Example 5: Decide if given three points are on the same line. [latex]A=(1,0,2)[/latex], [latex]B=(2,1,0)[/latex], [latex]C=(0,2,-1)[/latex].

Exercise 5: Decide if given three points are on the same line. [latex]A=(1,3,2)[/latex], [latex]B=(3,-1,6)[/latex], [latex]C=(5,2,0)[/latex].

Example 6: Decide if given two lines are parallel to each others.

\begin{align*}L_{1}: & \frac{x-2}{3}=\frac{y+1}{2}=\frac{z-1}{-2}\\L_{2}: & \frac{x+1}{9}=\frac{y-1}{6}=\frac{z+1}{-4}\end{align*}

Exercise 6: Decide if given two lines are parallel to each others.

\begin{align*} L_{1}: & \frac{x+2}{3}=\frac{y-1}{1}=\frac{z-1}{-2}\\ L_{2}: & \frac{x-1}{6}=\frac{y}{3}=\frac{z+1}{-6} \end{align*}

Example 7: Decide if given two lines intersect to each others.

\begin{align*} L_{1}: & \frac{x-3}{2}=\frac{y}{1}=\frac{z-2}{1}\\ L_{2}: & \frac{x}{1}=\frac{y}{-1}=\frac{z}{1} \end{align*}

Exercise 7: Decide if given two lines intersect to each others.

\begin{align*} L_{1}: & \frac{x-2}{1}=\frac{y+1}{1}=\frac{z-1}{-2}\\ L_{2}: & \frac{x+1}{-3}=\frac{y+2}{-1}=\frac{z-2}{1} \end{align*}

Given any two lines, [latex]L_{1}[/latex] and [latex]L_{2}[/latex], in the space, there are four possible for the relationships of those two lines. They are equal or parallel but not equal, or intersecting at exactly one point or skew which means they are not parallel nor intersecting to each others.

Example 8: Determine whether the lines are equal, parallel but not equal, skew, or intersecting.

\begin{align*}  L_{1}: & \frac{x-3}{2}=\frac{y}{1}=\frac{z-2}{1}\\ L_{2}: & \frac{x}{1}=\frac{y}{-2}=\frac{z}{1} \end{align*}

Group work:

1. Determine whether the lines are equal, parallel but not equal, skew, or intersecting. [latex]L_{1}:x=y-1=-z[/latex] and [latex]L_{2}:x-2=-y=\frac{z}{2}[/latex].

2. Determine whether the lines are equal, parallel but not equal, skew, or intersecting. [latex]L_{1}:x=2t,y=0,z=3,\text{ where }t\in R[/latex] and [latex]L_{2}:x=0,y=8+s,z=7+s,\text{ where }s\in R[/latex].

3. Determine whether the lines are equal, parallel but not equal, skew, or intersecting. [latex]L_{1}:x=-1+2t,y=1+3t,z=7t,\text{ where }t\in R[/latex] and [latex]L_{2}:x-1=\frac{2}{3}(y-4)=\frac{2}{7}z-2[/latex].

4. Consider line [latex]L[/latex] of symmetric equations [latex]x-2=-y=\frac{z}{2}[/latex] and point [latex]A(1,1,1)[/latex].

a. Find parametric equations for a line parallel to [latex]L[/latex] that passes through point [latex]A[/latex].

b. Find symmetric equations of a line skew to [latex]L[/latex] and that passes through point [latex]A[/latex].

c. Find symmetric equations of a line that intersects [latex]L[/latex] and passes through point [latex]A[/latex].