To find the equation of a line in the -plane, we need two pieces of information: a point and the slope. The slope conveys direction information. As vertical lines have an undefined slope, the following statement is more accurate:
Let be a point in space, let be the vector with initial point at the origin and terminal point at (i.e., “points” to ), and let be a vector. Consider the points on the line through in the direction of .
Clearly one point on the line is ; we can say that the vector lies at this point on the line. To find another point on the line, we can start at and move in a direction parallel to . For instance, starting at and traveling one length of places one at another point on the line. Consider Figure 11.5.2 where certain points along the line are indicated.
The figure illustrates how every point on the line can be obtained by starting with and moving a certain distance in the direction of . That is, we can define the line as a function of :
On the left is the equation for a line in the plane with slope and intercept . On the right is the equation for a line in space through the point with direction vector .
Above the two equations is the text “Starting Point”. From this text are two arrows, pointing to the value in the plane equation, and the value in the space equation. Also above the equations is the text “Direction”. From this text, two arrows point to the value in the plane equation, and the value in the space equation.
Below the two equations is the text “How Far To Go In That Direction”. Arrows point from this text to the value in the plane equation, and the value in the space equation.
Figure11.5.3.Understanding the vector equation of a line
Equation (11.5.1) is an example of a vector-valued function; the input of the function is a real number and the output is a vector. We will cover vector-valued functions extensively in the next chapter.
The last line states that the values of the line are given by , the values are given by , and the values are given by . These three equations, taken together, are the parametric equations of the line through in the direction of .
assuming . Since is equal to each expression on the right, we can set these equal to each other, forming the symmetric equations of the line through in the direction of :
Each representation has its own advantages, depending on the context. We summarize these three forms in the following definition, then give examples of their use.
The first two equations of the line are useful when a value is given: one can immediately find the corresponding point on the line. These forms are good when calculating with a computer; most software programs easily handle equations in these formats. (For instance, the graphics program that made Figure 11.5.6 can be given the input “(2-t,3+t,1+2*t)” for .).
Does the point lie on the line? The graph in Figure 11.5.6 makes it clear that it does not. We can answer this question without the graph using any of the three equation forms. Of the three, the symmetric equations are probably best suited for this task. Simply plug in the values of , and and see if equality is maintained:
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We see that does not lie on the line as it did not satisfy the symmetric equations.
Recall the statement made at the beginning of this section: to find the equation of a line, we need a point and a direction. We have two points; either one will suffice. The direction of the line can be found by the vector with initial point and terminal point :.
The parametric equations of the line through in the direction of are:
A graph of the points and line are given in Figure 11.5.8. Note how in the given parametrization of the line, corresponds to the point , and corresponds to the point . This relates to the understanding of the vector equation of a line described in Figure 11.5.3. The parametric equations “start” at the point , and determines how far in the direction of to travel. When , we travel 0 lengths of ; when , we travel one length of , resulting in the point .
Subsection11.5.2Parallel, Intersecting and Skew Lines
In the plane, two distinct lines can either be parallel or they will intersect at exactly one point. In space, given equations of two lines, it can sometimes be difficult to tell whether the lines are distinct or not (i.e., the same line can be represented in different ways). Given lines and , we have four possibilities: and are
We start by looking at the directions of each line. Line has the direction given by and line has the direction given by . It should be clear that and are not parallel, hence and are not the same line, nor are they parallel. Figure 11.5.10 verifies this fact (where the points and directions indicated by the equations of each line are identified).
We next check to see if they intersect (if they do not, they are skew lines). To find if they intersect, we look for and values such that the respective , and values are the same. That is, we want and such that:
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This is a relatively simple system of linear equations. Since the last equation is already solved for , substitute that value of into the equation above it:
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A key to remember is that we have three equations; we need to check if satisfies the first equation as well:
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It does not. Therefore, we conclude that the lines and are skew.
It is obviously very difficult to simply look at these equations and discern anything. This is done intentionally. In the “real world,” most equations that are used do not have nice, integer coefficients. Rather, there are lots of digits after the decimal and the equations can look “messy.”
We again start by deciding whether or not each line has the same direction. The direction of is given by and the direction of is given by . When it is not clear through observation whether two vectors are parallel or not, the standard way of determining this is by comparing their respective unit vectors. Using a calculator, we find:
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The two vectors seem to be parallel (at least, their components are equal to 4 decimal places). In most situations, it would suffice to conclude that the lines are at least parallel, if not the same. One way to be sure is to rewrite and in terms of fractions, not decimals. We have
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One can then find the magnitudes of each vector in terms of fractions, then compute the unit vectors likewise. After a lot of manual arithmetic (or after briefly using a computer algebra system), one finds that
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We can now say without equivocation that these lines are parallel.
Are they the same line? The parametric equations for a line describe one point that lies on the line, so we know that the point lies on . To determine if this point also lies on , plug in the , and values of into the symmetric equations for :
The point lies on both lines, so we conclude they are the same line, just parametrized differently. Figure 11.5.12 graphs this line along with the points and vectors described by the parametric equations. Note how and are parallel, though point in opposite directions (as indicated by their unit vectors above).
Given a point and a line in space, it is often useful to know the distance from the point to the line. (Here we use the standard definition of “distance,” i.e., the length of the shortest line segment from the point to the line.) Identifying with the point ,Figure 11.5.13 will help establish a general method of computing this distance .
A generic line is shown, without coordinates. On the line is a point and a direction vector . Near the line is a point . The vector points from the point on the line to the point off the line. This vector forms the hypotenuse of a right-angled triangle whose base is part of the line, and whose height, , is shown as a dashed line segment from to the line. The angle between and is also shown. The height is the perpendicular distance from to the line.
Figure11.5.13.Establishing the distance from a point to a line
It is also useful to determine the distance between lines, which we define as the length of the shortest line segment that connects the two lines (an argument from geometry shows that this line segments is perpendicular to both lines). Let lines and be given, as shown in Figure 11.5.15. To find the direction orthogonal to both and , we take the cross product: . The magnitude of the orthogonal projection of onto is the distance we seek:
Let be a point on a line that is parallel to . The distance from a point to the line is:
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Let be a point on line that is parallel to , and let be a point on line parallel to , and let , where lines and are not parallel. The distance between the two lines is:
The equation of the line gives us the point that lies on the line, hence . The equation also gives . Following Key Idea 11.5.17, we have the distance as
These are the sames lines as given in Example 11.5.9, where we showed them to be skew. The equations allow us to identify the following points and vectors:
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From Key Idea 11.5.17 we have the distance between the two lines is
One of the key points to understand from this section is this: to describe a line, we need a point and a direction. Whenever a problem is posed concerning a line, one needs to take whatever information is offered and glean point and direction information. Many questions can be asked (and are asked in the Exercise section) whose answer immediately follows from this understanding.
Lines are one of two fundamental objects of study in space. The other fundamental object is the plane, which we study in detail in the next section. Many complex three dimensional objects are studied by approximating their surfaces with lines and planes.