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Line of Intersection of Two Plane and Angle Between a Line and a Plane, Intersection of Line and Plane is considered one of the most asked concept.
31 Questions around this concept.
The distance of the point (1, 0, 2) from the line point of intersection of the line
and the plane
, is:
If the line, lies in
the plane, then
is equal to:
The intersection of two non-parallel planes always form a line, for example in three dimensional coordinate system, intersection of the XY plane with XZ plane forms X-axis.
What is the equation of line when YZ plane and XZ plane intersects?
Let the equation of two non-parallel planes be and
.
Now, line of intersection of planes is perpendicular to and
.
Therefore, line of intersection is parallel to vector .
Hence the vector equation for the line of intersection is given by
Where, is the vector result of the normal vector of the two planes.
To find the line of intersection of planes a1x+b1y+c1z=d1 and a2x+b2y+c2z=d2, then first find any point on the line by putting z = 0 (say), then we can find corresponding values of x and y be solving equations a1x+b1y+c1z=d1 and a2x+b2y+c2z=d2. Thus, by fixing the value of z = λ we can find the corresponding value of x and y in terms of λ. After getting x, y and z in terms of λ we can find the equation of line in symmetric form.
Illustration
The line of intersection of two given plane P1: -3x + 2y - 3z - 1 = 0 and P2: 2x - y - 4z + 2 = 0 is
Let z = λ
Then, -3x + 2y = 1 + 3λ
and 2x - y = -2 + 4λ
Solve these two equations, x = -3 + 11λ and y = -4 + 18λ
The equation of the line is
Aliter
The general equation of plane and its normal is ax + by + cz + d = 0 and
Then,
and
To write the equation of the line of intersection, i.e., , we still need the coordinates of any of its point P(x0, y0, z0).
Let this point be the intersection of the intersection line and the XY coordinate plane.
Then, the coordinates of the point of intersection (x, y, 0) must satisfy equations of the given planes.
Therefore, by putting z = 0 into P1 and P2 we get,
-3x + 2y - 3(0) - 1 = 0
2x - y - 4(0) + 2 = 0
x = -3 and y = -4
So the line of intersection is .
Angle Between a Line and a Plane
The angle between a line and a plane is the complement of the angle between the line and normal to the plane.
Vector Form
If the equation of the line is and the equation of the plane is
. Then the angle θ between the line and the normal to the plane is
and so the angle φ between the line and the plane is given by 900 – θ,
Cartesian Form
The angle between a line and a plane
If the line is
and the plane is is given by
NOTE:
Line and plane
are perpendicular if
or
and parallel if
or
Given equation of the line is and the equation of the plane is ax + by + cz + d = 0
Put the value of r in , you will get coordinates of point P.
Condition for a Line to be Parallel to a Plane
The line is parallel to plane ax + by + cz + d = 0 iff:
Condition for a Line to Lie in the Plane
Condition for the line to lie in the plane ax + by + cz + d = 0 are:
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