IB AA: Vector Lines and Planes

Moving from two-dimensional coordinate geometry into three-dimensional space is a pivotal step in mathematics. It transforms abstract algebra into a powerful language for describing the real world—from the flight path of a drone to the orientation of a crystal face. Mastering vectors for lines and planes provides the essential toolkit for modeling and solving spatial problems, forming a cornerstone of the IB Analysis & Approaches syllabus.

Lines in 3D Space

In two dimensions, a line has a unique slope-intercept form. In three dimensions, this idea generalizes beautifully using vectors. The most powerful way to define a line in 3D is with a vector equation. You need just two pieces of information: a point on the line and its direction.

Given a point with position vector $a$ and a direction vector $b$, the vector equation of the line is $r=a+\lambda b$, where $\lambda$ is a scalar parameter and $r$ is the position vector of any point on the line. This equation tells you: "Start at point $a$, and travel any distance (dictated by $\lambda$) along the direction $b$."

From this, you can derive the parametric equations. If $a=(x_0,y_0,z_0)$ and $b=(l,m,n)$, then the coordinates of any point are given by: $$x=x_0+\lambda l,y=y_0+\lambda m,z=z_0+\lambda n.$$ A related form is the Cartesian (or symmetric) form, obtained by solving each parametric equation for $\lambda$ and equating them: $\frac{x-x_0}{l}=\frac{y-y_0}{m}=\frac{z-z_0}{n}$, provided $l,m,$ and $n$ are all non-zero.

Relative Positions of Two Lines

In space, two lines can be parallel, intersecting, or skew. To determine which, you compare their direction vectors and check for a common point.

If the direction vectors are scalar multiples, the lines are parallel (or coincident if they share a point). If they are not parallel, set their parametric equations equal to solve for the parameters $\lambda$ and $\mu$. If you find a consistent $\lambda$ and $\mu$ that satisfy all three coordinate equations, the lines intersect at a unique point. If the direction vectors are not parallel and no consistent point solution exists, the lines are skew. Skew lines are the unique 3D phenomenon: they are not parallel and never meet, yet they are not in the same plane. Imagine one line running along the ceiling of a room and another along the floor in a perpendicular direction—they don't intersect but aren't parallel.

Planes in 3D Space

While a line is defined by a point and a direction, a plane is defined by a point and a direction perpendicular to it, called a normal vector, $n$. Think of the normal vector as an arrow sticking straight out of a flat sheet.

The standard equation arises from the geometric fact that for any point $r$ in the plane, the vector $(r-a)$ is perpendicular to $n$. Their scalar product is zero: $n\cdot (r-a)=0$. This expands to the scalar product form: $n\cdot r=n\cdot a$.

If $n=(a,b,c)$ and the constant $n\cdot a=d$, you get the Cartesian form of a plane: $ax+by+cz=d$. Here, $(a,b,c)$ is the normal vector. Other useful forms include the vector equation using two non-parallel direction vectors in the plane: $r=a+\lambda b+\mu c$.

Intersections Involving Planes

Finding where a line intersects a plane is a common problem. You substitute the parametric equations of the line ($x=x_0+\lambda l$, etc.) into the Cartesian equation of the plane ($ax+by+cz=d$). This yields an equation in the single variable $\lambda$. Solve for $\lambda$, then substitute back into the line's equations to find the coordinates of the intersection point. If the equation for $\lambda$ is inconsistent (e.g., $0=5$), the line is parallel to the plane and does not intersect it. If it simplifies to an identity (e.g., $0=0$), the entire line lies within the plane.

The intersection of two planes is typically a line. To find its equation, you solve their Cartesian equations simultaneously. Since you have two equations with three unknowns, you will have one free parameter, which defines the line. A systematic method is to let one variable (e.g., $z$) equal a parameter $\lambda$, then solve for $x$ and $y$ in terms of $\lambda$. If the planes are parallel (their normal vectors are scalar multiples), they are either coincident or distinct and never intersect.

Angles in Space

Vector geometry provides an elegant method for calculating angles between any two objects: you find the angle between their key direction vectors.

For the angle between two lines, use their direction vectors, $b_1$ and $b_2$. The angle $\theta$ is given by: $$\cos \theta =\frac{|b_1\cdot b_2|}{|b_1||b_2|}.$$ We take the absolute value to ensure the acute angle is found.

For the angle between a line and a plane, first find the angle $\varphi$ between the line's direction vector $b$ and the plane's normal vector $n$, using $\cos \varphi =\frac{|b\cdot n|}{|b||n|}$. The angle $\theta$ between the line and the plane is the complement: $\theta =90^{\circ }-\varphi$ (or $\frac{\pi }{2}-\varphi$ in radians). In other words, $\sin \theta =\frac{|b\cdot n|}{|b||n|}$.

For the angle between two planes, simply find the angle between their normal vectors, $n_1$ and $n_2$, using $\cos \theta =\frac{|n_1\cdot n_2|}{|n_1||n_2|}$. This gives the acute dihedral angle between the planes.

Common Pitfalls

1. Misidentifying Skew Lines: A common error is to assume non-parallel lines in 3D must intersect. Always perform the full check: confirm direction vectors are not parallel, then attempt to solve for an intersection point. Failure to find one confirms the lines are skew.
2. Incorrect Normal Vector from a Cartesian Equation: In the plane equation $ax+by+cz=d$, the normal vector is $(a,b,c)$. Students sometimes mistakenly include the constant $d$. Remember, $d$ is a scalar result of $n\cdot a$, not a component of the direction.
3. Angle Formula Confusion: Mixing up the formulas for line-line and line-plane angles is frequent. Use the cosine with the absolute value of the dot product for angles between lines or planes. For the line-plane angle, you must use the sine, as it involves the complement of the angle to the normal.
4. Parameter Mishandling in Intersections: When finding where a line meets a plane, substitute the entire parametric expression into the plane equation. A mistake is to substitute just the base point $a$ instead of the general point $a+\lambda b$.

Summary

• The vector equation $r=a+\lambda b$ is the most fundamental description of a line in 3D, easily converted to parametric or Cartesian forms.
• Two lines in space can be parallel, intersecting, or skew; skew lines are non-parallel, non-intersecting, and exist only in three or more dimensions.
• A plane is defined by a point and a normal vector $n$, leading to the key forms $n\cdot r=n\cdot a$ and $ax+by+cz=d$.
• The intersection of a line and a plane is found by substitution, while the intersection of two planes is typically a line found by solving the two equations simultaneously.
• Angles are calculated using dot products: between lines or planes, use $\cos \theta$; between a line and a plane, use $\sin \theta =\frac{|b\cdot n|}{|b||n|}$.