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Maths Exercices

Line-Plane Angle


The angle between a line, r, and a plane, π, is the angle between r and its orthogonal projection onto π, r'.

That is, the angle between a line and a plane is equal to the complementary acute angle that forms between the direction vector of the line and the normal vector of the plane.

Let's name θ to the angle which line makes with plane, u=(u1,u2,u3) the direction vector of line and n=(A,B,C) the normal vector of the plane.


Determine the angle between the line:



and the plane

Solution:

u=(2,1,2) and n=(1,1,0)

Therefore


θ=45º


Determine the angle between the line

x+3y-z+3=0
2x-y-z-1=0

and the plane

Solution:

u=
i j k
1 3 -1
2 -1 -1
=-4i-j-7k u=(-4,-1,-7)


n=(2,-1,3)

θ=22.91º