(i) If the line is perpendicular to the plane, then the line is parallel to the normal to the plane.
(ii) If the line is parallel to the plane, then the line is perpendicular to the normal to the plane.

**Angle
between a line and a plane**

We know that the angle between a line and a plane is the complement of the
angle between the normal to the plane and the line

Let = + t be the equation of the line and â‹… = p be the equation of the plane. We know that * *is
parallel to the given line and * *is normal to the given plane.
If *Î¸ *is the acute angle between the line and the plane, then the acute
angle between * *and * *is ((Ï€/2)-Î¸).Therefore,

So, the acute angle between the line and the plane is given by Î¸ = â€¦.(1)

In Cartesian form if and *ax* + *by* + *cz* = *p*
are the equations of the line and the plane, then = *a*_{1}*i*Ë† + *b*_{1} Ë†*j* + c_{1}*k*Ë† and = aË†*i*
+ bË†*j* + cË†*k* . Therefore, using (1), the acute angle Î¸ between the line and
plane is given by

**Remark**

(i) If the line is perpendicular to the plane, then the line is
parallel to the normal to the plane.

So, is perpendicular to . Then we have = Î». Î» âˆˆ **R** ,which gives

(ii) If the line is parallel to the plane, then the line is
perpendicular to the normal to the plane.

Therefore, . = 0 â‡’ *aa*_{1} + *bb _{1}*
+ cc

**Example 6.48**

Find the angle between the straight line = (2Ë†*i* + 3Ë†*j* + Ë†*k* )+ *t* (Ë†*i*
- Ë†*j* + Ë†*k* ) and the plane 2x - y + z = 5 .

**Solution**

The angle between a line = + t and a plane â‹… = p with normal is Î¸

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

12th Mathematics : UNIT 6 : Applications of Vector Algebra : Angle between a line and a plane |

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