Angle between a Line and a Plane (Cartesian Form)

In 3D geometry, understanding the angle between a line and a plane is crucial for solving various problems related to spatial figures. This topic is often covered in advanced mathematics courses and is essential for students preparing for exams.

Cartesian Form

Before we delve into the angle between a line and a plane, let's briefly review the Cartesian forms for the equation of a line and a plane.

Equation of a Line

A line in 3D space can be represented in Cartesian form as:

$$ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} $$

where $(x_1, y_1, z_1)$ is a point on the line, and $a, b, c$ are the direction ratios of the line.

Equation of a Plane

A plane in 3D space can be represented in Cartesian form as:

$$ Ax + By + Cz + D = 0 $$

where $A, B, C$ are the direction ratios of the normal to the plane, and $D$ is the distance from the origin to the plane along the normal.

Angle between a Line and a Plane

The angle between a line and a plane is defined as the complement of the angle between the line and the normal to the plane. If a line is parallel to the plane, the angle is 0 degrees, and if it is perpendicular to the plane, the angle is 90 degrees.

Formula

The angle $\theta$ between a line with direction ratios $a, b, c$ and a plane with a normal having direction ratios $A, B, C$ is given by:

$$ \cos(\theta) = \frac{|aA + bB + cC|}{\sqrt{a^2 + b^2 + c^2} \cdot \sqrt{A^2 + B^2 + C^2}} $$

Here, $\theta$ is the angle between the line and the normal to the plane. To find the angle between the line and the plane itself, we calculate the complement:

$$ \text{Angle between line and plane} = 90^\circ - \theta $$

Important Points

Point	Description
Angle with Normal	The angle between the line and the normal to the plane is directly calculated using the dot product formula.
Complementary Angle	The actual angle between the line and the plane is the complement of the angle with the normal.
Perpendicularity	If the line is perpendicular to the plane, the angle between them is 0 degrees.
Parallelism	If the line is parallel to the plane, the angle between them is 90 degrees.

Examples

Example 1: Finding the Angle

Given a line with direction ratios $1, -2, 2$ and a plane with a normal having direction ratios $3, 2, -6$, find the angle between the line and the plane.

Solution:

First, we find the angle between the line and the normal to the plane using the dot product formula:

$$ \cos(\theta) = \frac{|1 \cdot 3 + (-2) \cdot 2 + 2 \cdot (-6)|}{\sqrt{1^2 + (-2)^2 + 2^2} \cdot \sqrt{3^2 + 2^2 + (-6)^2}} $$

$$ \cos(\theta) = \frac{|3 - 4 - 12|}{\sqrt{1 + 4 + 4} \cdot \sqrt{9 + 4 + 36}} $$

$$ \cos(\theta) = \frac{|-13|}{\sqrt{9} \cdot \sqrt{49}} $$

$$ \cos(\theta) = \frac{13}{3 \cdot 7} $$

$$ \cos(\theta) = \frac{13}{21} $$

Now, we find the angle $\theta$:

$$ \theta = \cos^{-1}\left(\frac{13}{21}\right) $$

And the angle between the line and the plane is the complement:

$$ \text{Angle between line and plane} = 90^\circ - \theta $$

Example 2: Line Parallel to Plane

Consider a line with direction ratios $2, 3, 4$ and a plane with a normal having direction ratios $1, -2, 2$. If the dot product of the direction ratios of the line and the normal to the plane is zero, the line is parallel to the plane.

Solution:

Calculate the dot product:

$$ 2 \cdot 1 + 3 \cdot (-2) + 4 \cdot 2 = 2 - 6 + 8 = 4 $$

Since the dot product is not zero, the line is not parallel to the plane. To find the angle, we would use the formula as shown in Example 1.

By understanding the Cartesian forms and the formula for the angle between a line and a plane, students can solve a variety of problems in 3D geometry. Practice with different sets of direction ratios and equations will help solidify this concept for exams.