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# Meeting Point of a Line and a Plane

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

Meeting Point of a Line and a Plane

### Theorem 6.23

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

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  = p for some value of t , say t1. So, We get Example 6.56

Find the coordinates of the point where the straight line intersects the plane x y + z − 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, −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 t = 0.

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

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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