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 $Pi$