Question #b2bfd

2 Answers
Mar 15, 2017

Answer:

See below.

Explanation:

#x+y+z+3 =0# is the equation of a plane in #RR^3#. This plane can be generated by a linear form with the structure

#Pi->p = p_0+lambda_1 vec v_1+lambda_2 vec v_2#

The given plane is given with the structure

#Pi_0-> << p-p_0,vec v >> = 0#

with

#p = (x,y,z)#
#p_0=(-1,-1,-1)# and
#vec v = (1,1,1)#

Here #Pi# and #Pi_0# are equivalent if #vec v = vec v_1 xx vec v_2# because then

#Pi_0-> << p_0+lambda_1 vec v_1+lambda_2 vec v_2-p_0, vec v >> = 0#
The affine space #Pi# is generated by the linear combinations of two independent vectors #{vec v_1, vec v_2} in RR^3# with the property:

#vec v_1 ne 0, vec v_2 ne 0, << vec v_1, vec v >> = << vec v_2, vec v >>=0#

Mar 15, 2017

The linear algebra answer.

This matrix has 3 independent variables but is of rank 1 only , so it will have #3 - 1 = 2# linearly independent solutions.