Chapter 1: Vectors, Lines and Planes
Section 1.6: Lines
Table 1.6.1 lists the various ways a line can be defined in ℝ3.
x=a+t u, y=b+t v,z=c+t w
Table 1.6.1 Forms for a line in ℝ3
In the vector form, P is a position vector to the fixed point P that the line achieves when t=0. The vector V is the direction of the line.
The parametric form is obtained from the vector form by equating corresponding components.
The symmetric form is obtained from the parametric form by solving each equation for the parameter t. The parametric form is obtained from the symmetric form by setting each fraction equal to the same quantity, namely, t.
Obtain an equation for the line that passes through the point P:3,2,1 and that is parallel to the vector V=2 i−3 j+5 k.
Obtain an equation for the line through the points P:3,2,1 and Q:5,−1,4.
The lines R=A+t P and R=B+s Q, are defined respectively by the parametric equations
x=3−2 t,y=2+5 t,z=6+t and x=5+4 s,y=7+2 s,z=3+2 s
Show that these lines do not intersect and are not parallel (so they are skew lines).
Find the common normal between them.
Calculate the distance between them.
Find the distance from the point P:2,1,−3 to the line R=A+t V, where A=i−j+2 k and V=7 i−9 j+5 k.
Lines L1 and L2 both have the common direction V=3 i−2 j+5 k, with L1 passing through point P:7,−4,6 and L2 passing through Q:4,−7,1.
Find the equations of L1 and L2.
Calculate the distance between L1 and L2.
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