How do you write a vector equation and a parametric equation for each line: the line through A(-1,2,1) and B(1,2,1)?

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Answer 1 · 2016-10-27T01:18:01

Vector equation
#r(lambda)=(-1,2,1)+lambda(2,0,0)#

Parametric equation
#" "##x=-1+2lambda#

r(t) = #y=2#

#" "##z=1#

The vector equation of a line involves a point on the line and the vector direction of the line.

So using point A #(-1,2,1)# and direction #vec(AB)#

we get a line with vector equation

#r(lambda)=A+lambda*vec(AB)#

#r(lambda)=(-1,2,1)+lambda((1,2,1)-(-1,2,1))#

#r(lambda)=(-1,2,1)+lambda(2,0,0)#

To parametrically define this line we now take each component (x,y,z) or (i,j,k) individually

#" "##x=-1+2lambda#

r(t) = #y=2#

#" "##z=1#