Intersections of Lines and Planes
Line-Plane Intersection
- The intersection of a line and a plane in three-dimensional space is a fundamental concept in analytic geometry.
- This intersection can result in one of three outcomes:
- A single point of intersection
- No intersection (the line is parallel to the plane)
- The entire line (when the line lies entirely within the plane)
To find the point of intersection, we typically use a system of equations.Note
The geometric interpretation of this solution is crucial. It represents the unique point where the line pierces the plane in 3D space.
Derivation of the Angle Between a Line and a Plane
- The angle $\theta$ between a line and a plane is the complement of the angle between the direction vector of the line and the normal vector of the plane.
- To derive this formula, consider a line with direction vector $\mathbf{v}$ and a plane with normal vector $\mathbf{n}$.
- The angle $\phi$ between the line and the plane's normal vector can be calculated using the dot product formula:
$$ \phi = \frac{\mathbf{v} \cdot \mathbf{n}}{|\mathbf{v}||\mathbf{n}|}$$
- Since the angle $\theta$ between the line and the plane is the complement of $\phi$, we use:
$$\theta = 90^\circ - \phi$$
- Taking the sine of both sides and rearranging, we obtain:
$$\theta = \frac{|\mathbf{v} \cdot \mathbf{n}|}{|\mathbf{v}||\mathbf{n}|}$$
NoteThis formula shows that the shortest angle between a line and a plane depends only on their directional relationship and provides a straightforward way to compute it.
ExampleConsider a line given by the parametric equations:
$x = 1 + 2t$ $y = 3 - t$ $z = 2 + 3t$
And a plane given by the equation:
$2x + 3y - z = 4$
To find the intersection, we substitute the line equations into the plane equation:
$2(1 + 2t) + 3(3 - t) - (2 + 3t) = 4$
Simplifying:
$2 + 4t + 9 - 3t - 2 - 3t = 4$ $9 - 2t = 4$ $-2t = -5$ $t = \frac{5}{2}$
Substituting this value of $t$ back into the line equations gives us the point of intersection:
$x = 1 + 2(\frac{5}{2}) = 6$ $y = 3 - \frac{5}{2} = \frac{1}{2}$ $z = 2 + 3(\frac{5}{2}) = \frac{17}{2}$
Therefore, the point of intersection is $(6, \frac{1}{2}, \frac{17}{2})$.
Intersection of Two Planes
When two planes intersect, they typically form a line. However, there are two special cases:
- The planes are parallel and don't intersect
- The planes are identical and overlap completely
To find the line of intersection, we solve the equations of the two planes simultaneously.
