Intersections and angles involving planes (HL)

Introduction: Navigating the 3D World

Welcome to the final frontier of the Vectors chapter! So far, you have learned how to describe lines and planes using equations. Now, we are going to look at how these objects interact in 3D space. Think of this as "3D geometry collisions." Whether you are designing a sleek modern building or programming a flight path for a drone, understanding where a line hits a wall (plane) or how two walls meet at an angle is essential. Don't worry if this seems abstract at first—we will break it down into simple, step-by-step processes.

1. Angles in 3D Space

The secret to finding angles involving planes is always to look at the normal vector \( \mathbf{n} \). Since the normal vector is the "representative" of the plane's orientation, it does most of the heavy lifting for us.

Angle Between a Line and a Plane

Imagine a flagpole (the line) stuck into the ground (the plane). To find the angle \( \theta \) between the line and the plane, we use the line’s direction vector \( \mathbf{v} \) and the plane’s normal vector \( \mathbf{n} \).

The "Sin" Twist: Normally, the scalar product uses \( \cos \). However, because the normal vector is perpendicular to the plane, the angle between the line and the normal is actually the complement of the angle we want. To skip the extra subtraction step, we use the sine formula:

\( \sin \theta = \frac{|\mathbf{v} \cdot \mathbf{n}|}{|\mathbf{v}| |\mathbf{n}|} \)

Quick Tip: If you accidentally use \( \cos \), you will find the angle between the line and the normal. Just subtract that from \( 90^\circ \) (or \( \frac{\pi}{2} \)) to get the correct answer!

Angle Between Two Planes

The angle between two planes is simply the angle between their two normal vectors, \( \mathbf{n_1} \) and \( \mathbf{n_2} \). It’s like measuring the angle of a folding book by looking at the spines.

\( \cos \theta = \frac{|\mathbf{n_1} \cdot \mathbf{n_2}|}{|\mathbf{n_1}| |\mathbf{n_2}|} \)

Key Takeaway: Use \( \sin \) for Line-Plane angles and \( \cos \) for Plane-Plane angles. Always use absolute value bars in the numerator to ensure you get the acute angle (the smaller one), which is the standard IB requirement unless stated otherwise.

2. Intersection of a Line and a Plane

When a line meets a plane, there are three possibilities: it hits at a single point, it never hits (parallel), or it lies entirely inside the plane.

Step-by-Step: Finding the Point of Intersection

1. Write the equation of the line in parametric form:
\( x = a_1 + \lambda d_1 \)
\( y = a_2 + \lambda d_2 \)
\( z = a_3 + \lambda d_3 \)

2. Substitute these expressions for \( x \), \( y \), and \( z \) into the Cartesian equation of the plane: \( ax + by + cz = d \).

3. Solve the resulting linear equation for the parameter \( \lambda \).

4. Plug your value of \( \lambda \) back into the line equation to find the specific coordinates \( (x, y, z) \).

What if things go wrong?
- If you get a statement like \( 5 = 5 \), the line lies entirely inside the plane (infinite intersections).
- If you get a statement like \( 0 = 7 \), the line is parallel to the plane and never hits it.

3. Intersection of Two Planes

When two non-parallel planes meet, they don't just hit at a point—they meet along a straight line. Think of the crease in the middle of an open book.

To find the equation of this line, you are essentially solving a system of two equations with three variables (\( x, y, z \)). Since there are more variables than equations, you will have a free parameter (usually \( \lambda \)).

How to solve:

1. Set one variable (like \( z \)) to be \( \lambda \).
2. Solve the remaining two equations for \( x \) and \( y \) in terms of \( \lambda \).
3. This gives you the parametric equation of the line of intersection.

Alternative Method: The direction of the intersection line is perpendicular to both normal vectors. You can find the direction vector \( \mathbf{v} \) by calculating the vector product (cross product): \( \mathbf{v} = \mathbf{n_1} \times \mathbf{n_2} \).

4. Intersection of Three Planes

This is where it gets interesting! This topic connects directly to Topic 1.16 (Systems of Linear Equations). When you have three planes, several things can happen:

1. A Unique Point: The planes meet like the corner of a room. This happens when the system has one unique solution.

2. A Line: The planes meet like the pages of a book sharing a single spine. This happens when there are infinitely many solutions (dependent equations).

3. No Intersection: There are a few ways this can happen:
- Two or more planes are parallel.
- The planes form a "triangular prism" shape where they meet in three separate parallel lines, but never all three at once.

Calculator Hint: For Paper 2 or 3, use your GDC's Poly Roots/System Solver tool to find these intersections quickly. If the calculator gives an error or a row of zeros, you know you have a "no solution" or "infinitely many solutions" case!

Common Mistakes to Avoid

- Mixing up Sine and Cosine: Remember: Line + Plane = \( \sin \). Plane + Plane = \( \cos \).
- Forgetting the Absolute Value: IB examiners usually want the acute angle. If your calculation gives a negative number for the dot product, the absolute value makes it positive so you get an angle between \( 0^\circ \) and \( 90^\circ \).
- Calculation Errors in Substitution: When plugging the line into the plane, be very careful with brackets and signs, especially with negative coordinates.

Key Takeaways Summary

- Line-Plane Angle: \( \sin \theta = \frac{|\mathbf{v} \cdot \mathbf{n}|}{|\mathbf{v}| |\mathbf{n}|} \)
- Plane-Plane Angle: \( \cos \theta = \frac{|\mathbf{n_1} \cdot \mathbf{n_2}|}{|\mathbf{n_1}| |\mathbf{n_2}|} \)
- Line-Plane Intersection: Substitute the line's parametric equations into the plane's Cartesian equation.
- Three Planes: Relates to systems of equations; can result in a point, a line, or no intersection at all.