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#

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Answer 1 · 2017-04-16T01:04:33

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 case 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 case 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 the present case 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#