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.
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).
Vector Equation of a Line in Space
By equating corresponding components, you can obtain parametric equations of a line in the space.
Parametric Equations of a Line in Space
x = x1+at
y = y1+bt
z = z1+ct
Symmetric Equations of a Line in Space
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>
x1=1, y1=-2, and z1=4
and direction numbers: a=2, b=4, and c=-4.
You will obtain:
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)
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)