Lines in the space
In the plane, slope is used to determine an equation of a line. In space it is more convenient to use vectors to determine the equation of a line.
In the following figure, consider the line L through the point P(x1,y1,z1) and parallel to the vector
v=. The vector v is a direction vector for the line L, and a, b, and c are direction numbers.
One way of describing the line L is to say that it consists of all points Q(x,y,z) for which the vector is parallel to v. This mean that is a scalar multiple of v, and you can write =vt, where t is a scalar (a real number).
= 1, y-y1, z-z1> = = tv
Vector Equation of a Line in Space
A line L parallel to the vector v = and passing through the point P(x1,y1,z1) is represented by the vector equation:
(x,y,z)=(x1,y1,z1)+t(a,b,c)
By equating corresponding components, you can obtain parametric equations of a line in the space.
Parametric Equations of a Line in Space
A line L parallel to the vector v = and passing through the point P(x1,y1,z1) is represented by the parametric equations:
x = x1+at
y = y1+bt
z = z1+ct
If the direction numbers a, b, and c are all nonzero, you can eliminate the parameter t to obtain symmetric equations of the line.
Symmetric Equations of a Line in Space
A line L parallel to the vector v = (where a, b, and c are all nonzero) and passing through the point P(x1,y1,z1) is represented by the symmetric equations:
Find parametric and symmetric equations of the line L that passes through the point (1,-2,4) and is parallel to v = <2,4,-4>
Solution:
To find a set of parametric equations of the line, use the coordinates
x1=1, y1=-2, and z1=4
and direction numbers: a=2, b=4, and c=-4.
You will obtain:
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x=1+2t
y=-2+4t
z=4-4t
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Parametric equations
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Because a, b, and c are all nonzero, a set of symetric equiations is:
Parametric Equations of a Line Through Two Points
Find a set of parametric equations of the line that passes through the points (-2,1,0) and (1,3,5)
Solution:
Begin by using the points P(-2,1,0) and Q(1,3,5) to find a direction vectro for the line passing through P and Q, given by:
v = = <1-(-2), 3-1, 5-0> = <3,2,5> =
Using the direction numbers a=3, b=2, and c=5 with the point P(-2,1,0), you can obtain the parametric equations:
x=-2+3t, y=1+2t and z=5t
Note:
As t varies over all real numbers, the parametric equations determine the points (x,y,z) on the line. In particular, note that t=0 and t=1 give the original points (-2,1,0) and (1,3,5)
