How do you write a vector equation and a parametric equation for each line: the line through A(1,-3,1) and parallel to vector u=(2,-2,1)?

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Answer 1 · 2017-01-04T16:48:52

vector eqn

#vecr=((1),(-3),(1))+lambda((2),(-2),(1))#

Parametric eqns

#x=1+2lambda#
#y=-3-2lambda#
#z=1+lambda#

the general vector eqn of a line

#vecr=veca+lambdavecd#

#vecr=#the general position vector on the line

#veca=# a known position vector on the line

#vecd=#the direction vector of the line.

In this case

#veca=((1),(-3),(1))#

#vecd=((2),(-2),(1))#

a vector eqn of the line is therefore

#vecr=((1),(-3),(1))+lambda((2),(-2),(1))#

For parametric eqns.

#vecr=((x),(y),(z))#

#((x),(y),(z))=((1),(-3),(1))+lambda((2),(-2),(1))#

#x=1+2lambda#
#y=-3-2lambda#
#z=1+lambda#

If you want the cartesian eqn. just eliminate #lambda#

#lambda=(x-1)/2#

#lambda=-(y+3)/2#

#lambda=z-1#

cartesian eqn

#(x-1)/2=-(y+3)/2=z-1#