Intersection of a Plane and a line
Parametric line equation L1: x = + t
 
y = + t
 
z = + t
 
Line equation L1: x +
= y +
= z +
 
Line defined
by 2 points
L1: x1 y1 z1
x2 y2 z2
Line defined
by vector
L1: Vector:      i + j + k
Point:     x:        y:        z:
Plane in three dimensional coordinates         Ax + By + Cz + D = 0
Plane equation:  x +  y +  z +   = 0
Plane passing through 3 points:  xa  ya  za
 xb  yb  zb
 xc  yc  zc
Plane and line intersection point:
 
Angle between plane and line:
 
                     
Intersection of plane and line equations
Plane and line intersection
Plane and line drawing
Plane:Ax + By + Cz + D = 0
Line:x = x1 + at
y = y1 + bt
z = z1 + ct
Note if the line is given by a vector
ai + bj + ck   and a point   (xp , yp , zp)
We can transalate to parametric form by:
x = xp + at
y = yp + bt
z = zp + ct
To find the intersection point P(x,y,z), substitute line parametric values of x, y and z into the plane equation:
A(x1 + at) + B(y1 + bt) + C(z1 + ct) + D = 0
and valuating t gives: t value
To find intersection coordinate substitute the value of t into the line equations:
Intersection point
Angle between the plane and the line:
Angle between plane and a line
Note: The angle is found by dot product of the plane vector and the line vector, the result is the angle between the line and the line perpendicular to the plane and θ is the complementary to π/2.
A line will be parallel to the plane if:           aA + bB + cC = 0
Example - Intersection of plane and line summary
Example:   Find the intersection point and the angle between the planes:    4x + z − 2 = 0    and the line
given in parametric form:       x =− 1 − 2t       y = 5       z = 1 + t
Solution:   Because the intersection point is common to the line and plane we can substitute the line parametric points into the plane equation to get:
4(− 1 − 2t) + (1 + t) − 2 = 0
t = − 5/7 = 0.71
Now we can substitute the value of   t   into the line parametric equation to get the intersection point.
x = − 1 − 2(− 5/7) = 3/7 = 0.43
y = 5
z = 1 − 5/7 = 2/7 = 0.29
And the intersection point is:         (0.43 , 5 , 0.29).
The angle between the line and the plane can be calculated by the cross product of the line vector with the vector representation of the plane which is perpendicular to the plane:     v = 4i + k
The line vector representation is the   t   portion of the parametric line equation:     n = -2i + k
Angle between plane and line
And the angle between the plane and the line is:         θ = π/2 − α = π/2 − 40.6 = 49.4 degree