For what value(s) of k, if any, will the line L: (3,4, −2) + t(−1, k, 3),t ∈ R be parallel to the plane 3x + y − 2z + 5 = 0?

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Answer 1 · 2017-05-19T15:12:25

#k=9#

Basically: a line parallel to a plane means that it is also perpendicular to the normal vector of the plane.

#L:vecr=(3,4,-2)+t(-1,k,3)#

we will put it in column vector form for easier handling.

#L:vecr=((3),(4),(-2))+t((-1),(k),(3))#

we will now rewrite the eqn of the plane in vector form

#3x+y-2z+5=0#

#=>Pi:((x),(y),(z))*((3),(1),(-2))+5=0#

#=>Pi:vecr*((3),(1),(-2))=-5#

now the standard eqn of a plane in vectors is:

#vecr*vecn=d#

where #" "vecn" "#is a normal vector to the plane

so if the line is parallel to this plane then the direction vector of the line and the normal vector of the plane are perpendicular.

ie#" "vecd*vecn=0#

we will now calculate this dot product to find #k#

#((-1),(k),(3))*((3),(1),(-2))=0#

#=>-3+k-6=0#

#k=9#