We observe that a straight line will lie in a plane if every point on the line, lie in the plane and the normal to the plane is perpendicular to the line.

**Condition
for a line to lie in a plane**

We observe that a straight line will lie in a plane if every
point on the line, lie in the plane and the normal to the plane is
perpendicular to the line.

i) If the line lies in the plane * *â‹…* *= *d *, then â‹… * *= *d and *. = 0

ii) if the line lies in the plane *Ax *+ *By *+ *Cz
*+ *D *= 0 , then

*Ax*_{1}
+ *By*_{1} + *Cz*_{1} + *D *= 0 and *aA *+ *bB *+ *cC *= 0

Verify whether the line lies in the plane 5*x *âˆ’ *y *+ *z *=
8 .

Here, ( *x*_{1}, *y*_{1}, *z*_{1}
) = (3, 4, âˆ’3) and direction ratios of the given straight line are (*a*,*b*,
*c*) = (âˆ’4, âˆ’7,12) . Direction ratios of the normal to the given plane are ( *A*,
*B*,*C *) = (5, âˆ’1,1) .

We observe that, the given point ( *x*_{1}, *y*_{1},
*z*_{1} ) = (3, 4, âˆ’3) satisfies the given plane 5*x *âˆ’ *y *+
*z *= 8

Next,* aA *+ *bB *+ *cC *= (âˆ’4)(5) + (âˆ’7)(âˆ’1) + (12)(1) = âˆ’1 â‰ 0
. So, the normal to the plane is not perpendicular to the line. Hence, the
given line does not lie in the plane.

Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail

12th Mathematics : UNIT 6 : Applications of Vector Algebra : Condition for a line to lie in a plane |

**Related Topics **

Privacy Policy, Terms and Conditions, DMCA Policy and Compliant

Copyright Â© 2018-2023 BrainKart.com; All Rights Reserved. Developed by Therithal info, Chennai.