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Angle between a line and a plane

The angle between a line, d, and a plane, π, is the angle between d and its orthogonal projection onto π, d’. Accordingly, the angle between a line and a plane can be considered equivalent to the complementary acute angle that forms between the direction vector of the line and the normal vector, of the plane.

From the formula of the scalar product of vectors find the cosine of the angle between the normal vector and direction vector we can conclude that:

Φ is considered as the angle between the line and the plane which is the complement of θ or 90 – θ as shown in the figure above. Recall that, cos θ = sin (90 – θ). So Φ can be given by:

sin (90 – θ) = cos θ

or

In the cartesian form the angle between this line and plane can be found using this formula:

If the line,, and the plane, π, are perpendicular, the direction vector of the line and the normal vector of the plane have the same direction and therefore its components are proportional:

Therefore:

Example:

Find the angle between the line and plane given below:

Line:

Plane: x – 2y + 3z + 4 = 0

Solution:Consider θ to be the angle between the line and the normal to the plane.

From the equation of the line, we find the direction vector as

From the equation of the plane, we find the normal vector as

Now we can use the equation discussed before to find the angle between the line and the plane:

Therefore, we can find the angle as:

Example: Determine the angle between the line and the plane given below:

Line:

Plane: x=1

Solution

We can also write line r as: :

Hence, we can obtain the direction vector as and normal vector

Now, we can substitute the values to obtain the angle between the direction vector and the normal vector:

Example: Find the coordinates of the point where the line through the points

A (3, 4, 1) and B(5, 1, 6) crosses through the XY-plane.

Solution: The vector equation of the line through points A and B is

Consider P to be the point where the line AB crosses through the XY-plane. Then the position

vector of the point P is of the form xi + yj. Hence we can write:

Now we equate to solve for x, y and z

Hence, the coordinates for the equations are:

 

 

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