The position vector of the point of intersection of the straight line and the plane

**Meeting
Point of a Line and a Plane**

The position vector of the point of intersection of the straight
line * *and the
plane

Let be the equation of the given line which is not parallel to the given plane whose equation is

Let * *be the position vector of the meeting
point of the line with the plane. Then * *satisfies both and * *â‹…* *= *p
*for some value of *t *, say *t*_{1}. So, We get

**Example 6.56**

Find the coordinates of the point where the straight line intersects the plane *x *âˆ’ *y *+ *z *âˆ’ 5 = 0 .

*Solution*

The vector form of the given plane is

We know that the position vector of the point of intersection of
the line and the plane

Therefore,the position vector of the point of intersection of the
given line and the given plane is

That is, the given straight line intersects the plane at the point
(2, âˆ’1, 2 ).

**Aliter**

The Cartesian equation of the given straight line is (say)

We know that any point on the given straight line is of the form
(3t + 2,4 t-1,2t+ 2) . If the given line and the plane intersects, then this
point lies on the given pane *x-y+z*-5=0.

So, (3t + 2) - (4t-1) + (2t+ 2) - 5 = 0 â‡’
t = 0.

Therefore, the given line intersects the given plane at the
point (2,-1, 2)

Tags : Definition, Theorem, Proof, Solved Example Problems, Solution , 12th Mathematics : UNIT 6 : Applications of Vector Algebra

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12th Mathematics : UNIT 6 : Applications of Vector Algebra : Meeting Point of a Line and a Plane | Definition, Theorem, Proof, Solved Example Problems, Solution

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