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 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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