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Parametric and symmetric equations of a line

A line L parallel to vector v = a , b , c and passing through point P ( x 0 , y 0 , z 0 ) can be described by the following parametric equations:

x = x 0 + t a , y = y 0 + t b , and z = z 0 + t c .

If the constants a , b , and c are all nonzero, then L can be described by the symmetric equation of the line:

x x 0 a = y y 0 b = z z 0 c .

The parametric equations of a line are not unique. Using a different parallel vector or a different point on the line leads to a different, equivalent representation. Each set of parametric equations leads to a related set of symmetric equations, so it follows that a symmetric equation of a line is not unique either.

Equations of a line in space

Find parametric and symmetric equations of the line passing through points ( 1 , 4 , −2 ) and ( −3 , 5 , 0 ) .

First, identify a vector parallel to the line:

v = −3 1 , 5 4 , 0 ( −2 ) = −4 , 1 , 2 .

Use either of the given points on the line to complete the parametric equations:

x = 1 4 t , y = 4 + t , and z = −2 + 2 t .

Solve each equation for t to create the symmetric equation of the line:

x 1 −4 = y 4 = z + 2 2 .
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Find parametric and symmetric equations of the line passing through points ( 1 , −3 , 2 ) and ( 5 , −2 , 8 ) .

Possible set of parametric equations: x = 1 + 4 t , y = −3 + t , z = 2 + 6 t ;

related set of symmetric equations: x 1 4 = y + 3 = z 2 6

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Sometimes we don’t want the equation of a whole line, just a line segment. In this case, we limit the values of our parameter t . For example, let P ( x 0 , y 0 , z 0 ) and Q ( x 1 , y 1 , z 1 ) be points on a line, and let p = x 0 , y 0 , z 0 and q = x 1 , y 1 , z 1 be the associated position vectors. In addition, let r = x , y , z . We want to find a vector equation for the line segment between P and Q . Using P as our known point on the line, and P Q = x 1 x 0 , y 1 y 0 , z 1 z 0 as the direction vector equation, [link] gives

r = p + t ( P Q ) .

Using properties of vectors, then

r = p + t ( P Q ) = x 0 , y 0 , z 0 + t x 1 x 0 , y 1 y 0 , z 1 z 0 = x 0 , y 0 , z 0 + t ( x 1 , y 1 , z 1 x 0 , y 0 , z 0 ) = x 0 , y 0 , z 0 + t x 1 , y 1 , z 1 t x 0 , y 0 , z 0 = ( 1 t ) x 0 , y 0 , z 0 + t x 1 , y 1 , z 1 = ( 1 t ) p + t q .

Thus, the vector equation of the line passing through P and Q is

r = ( 1 t ) p + t q .

Remember that we didn’t want the equation of the whole line, just the line segment between P and Q . Notice that when t = 0 , we have r = p , and when t = 1 , we have r = q . Therefore, the vector equation of the line segment between P and Q is

r = ( 1 t ) p + t q , 0 t 1 .

Going back to [link] , we can also find parametric equations for this line segment. We have

r = p + t ( P Q ) x , y , z = x 0 , y 0 , z 0 + t x 1 x 0 , y 1 y 0 , z 1 z 0 = x 0 + t ( x 1 x 0 ) , y 0 + t ( y 1 y 0 ) , z 0 + t ( z 1 z 0 ) .

Then, the parametric equations are

x = x 0 + t ( x 1 x 0 ) , y = y 0 + t ( y 1 y 0 ) , z = z 0 + t ( z 1 z 0 ) , 0 t 1 .

Parametric equations of a line segment

Find parametric equations of the line segment between the points P ( 2 , 1 , 4 ) and Q ( 3 , −1 , 3 ) .

By [link] , we have

x = x 0 + t ( x 1 x 0 ) , y = y 0 + t ( y 1 y 0 ) , z = z 0 + t ( z 1 z 0 ) , 0 t 1 .

Working with each component separately, we get

x = x 0 + t ( x 1 x 0 ) = 2 + t ( 3 2 ) = 2 + t ,
y = y 0 + t ( y 1 y 0 ) = 1 + t ( −1 1 ) = 1 2 t ,

and

z = z 0 + t ( z 1 z 0 ) = 4 + t ( 3 4 ) = 4 t .

Therefore, the parametric equations for the line segment are

x = 2 + t , y = 1 2 t , z = 4 t , 0 t 1 .
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Find parametric equations of the line segment between points P ( −1 , 3 , 6 ) and Q ( −8 , 2 , 4 ) .

x = −1 7 t , y = 3 t , z = 6 2 t , 0 t 1

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Distance between a point and a line

We already know how to calculate the distance between two points in space. We now expand this definition to describe the distance between a point and a line in space. Several real-world contexts exist when it is important to be able to calculate these distances. When building a home, for example, builders must consider “setback” requirements, when structures or fixtures have to be a certain distance from the property line. Air travel offers another example. Airlines are concerned about the distances between populated areas and proposed flight paths.

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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