How can I check if a given line lies in a given plane or not?

Question

For example, how do I check if the line #(1,2,3) + t(1,0,1)# lies in the plane# x+2y+2z+1=0#

Genesis Allen · Accepted Answer

See below.

A plane #Pi# can be represented as #<< p - p_0, vec n >> = 0# here #<< cdot, cdot >># represents the scalar product of two vectors. In our situation, we have #p = (x,y,z)# #vec n = (1,2,2)# and #p_0 = (-1,0,0)# A line #L# can be represented as #L->p = p_1+t vec v# In our situation, we have #p = (x,y,z)# #p_1=(1,2,3)# and #vec v = (1,0,1)# Now, if #L sub Pi# then #<< p_1+t vec v - p_0, vec n >> = 0, forall t in RR# or #<< p_1-p_0, vec n >> + t << vec v, vec n >> = 0# This occurs when #<< vec v, vec n >> = 0# being orthogonals, and also #<< p_1-p_0, vec n >> = 0# being orthogonals also. In this instance, we have #<< vec v, vec n >> = 1 xx 1+ 2xx0+2xx1=3 ne 0# and #<< p_1-p_0, vec n >> = 2xx1+ 2xx2+ 3xx2=12 ne 0# so concluding, #L# does not lies into #Pi#

Abigail Allen · Answer

undefined To check if a given line lies in a given plane or not, you can use the following method:

1. Find the parametric equations of the line.
2. Substitute the parametric equations into the equation of the plane.
3. If the resulting equation is satisfied, then the line lies in the plane. Otherwise, it does not.