How do you determine whether a line is parallel to a plane?

Linear Algebra question:





How do you determine whether a line is parallel to a plane?








How do I figure out whether this line is parallel to the following plane or not?





Line:


x = 3t


y = 1 +2t


z = 2-t





Plane:





4x-y+2z = 1





Thanks in advance.|||If the line and a normal vector of the plane are orthogonal then the line and the plane are parallel.





one normal vector to the plane is : v (4 ; - 1 ; 2)





one vector of the line is (3 ; 2 ; - 1)








scalar product of the two vectors =





(4 * 3) + ((- 1) * 2) + (2 * (- 1)) =





12 - 2 - 2 = 8





The result is not equal to 0 so the line and the plane are secant (not parallel).








Graph :





http://s636.photobucket.com/albums/uu88/鈥?/a>





*******|||If the directional vector of the line is orthogonal to the normal vector of the plane, the line and the plane are parallel. The dot product of orthogonal vectors is zero.





Let


v = directional vector of line


n = normal vector of plane





v = %26lt;3, 2, -1%26gt;


n = %26lt;4, -1, 2%26gt;





Take the dot product.


v 鈥?n = %26lt;3, 2, -1%26gt; 鈥?%26lt;4, -1, 2%26gt; = 12 - 2 - 2 = 8 鈮?0





The dot product is not zero, so the line and plane are not parallel.