Linear Algebra and Analytic Geometry · 120 h · Topic 5 of 9

Lines and planes

A point and a direction define a line; a point and a normal define a plane. Everything else follows from those two ideas.

Vector equation Normal vector Distances Angles Relative positions

01Two pieces of data are enough

All the analytic geometry of space rests on two simple ideas:

A line

It is defined by a point and a direction. The direction is given by a vector, called the direction vector.

A plane

It is defined by a point and a perpendicular. That perpendicular is given by the normal vector.

There are other ways to fix them too —two points determine a line; three non-collinear points, a plane—, but they all end up reducing to the two above. It is worth keeping in mind: almost all exercises are solved by finding the missing point and vector.

02Equations of the line

With a point P₀(x₀, y₀, z₀) and a direction vector d = (a, b, c), a generic point of the line is reached by walking from P₀ a multiple of d:

\[ \vec{r} = {\vec{r}}_0 + t \vec{d} \quad \begin{array}{c} x = x_0 + a t \\ y = y_0 + b t \\ z = z_0 + c t \end{array} \quad \frac{x -x_0}{a} = \frac{y -y_0}{b} = \frac{z -z_0}{c} \] Vector, parametric and symmetric equations: the same line written three ways. The parameter t covers the whole line as it goes from −∞ to +∞.

The symmetric equations fail if any component is zero

You cannot divide by a = 0. If the direction vector is (0, 2, 3), the line is written x = x₀ together with the other two fractions. The parametric equations do not have this problem, and that is why they are the ones to use when working.

In the plane, the same line also admits the slope-intercept form y = mx + h, the general form Ax + By + C = 0 and the intercept form x/p + y/q = 1, where p and q are the intercepts with the axes.

03Equations of the plane

If n = (A, B, C) is normal to the plane and P₀ belongs to it, then any point P of the plane satisfies that the vector P₀P is perpendicular to n. Writing that condition with the dot product gives everything:

\[ \vec{n} \cdot (\vec{r} -{\vec{r}}_0) = 0 \quad \implies \quad A x + B y + C z + D = 0 \] The coefficients of the general equation are the components of the normal. Reading the normal of a plane is reading the first three numbers.

If the plane is given by three points, the normal is obtained with a cross product: n = AB × AC. If it is given by two directions contained in it, the same. The cross product is the tool that goes from “directions inside the plane” to “direction perpendicular to it”.

Worked example

Plane through A(1, 0, 2), B(3, 1, 0) and C(0, 2, 1). AB = (2, 1, −2) and AC = (−1, 2, −1).

n = AB × AC = (1·(−1) − (−2)·2 , (−2)(−1) − 2(−1) , 2·2 − 1·(−1)) = (3, 4, 5).

With point A: 3(x − 1) + 4(y − 0) + 5(z − 2) = 0, that is 3x + 4y + 5z − 13 = 0. Check with B: 9 + 4 + 0 − 13 = 0 ✔ and with C: 0 + 8 + 5 − 13 = 0 ✔

04Relative positions

Between	What is compared	Cases
Two lines	The direction vectors and a point	Parallel (proportional direction vectors and no shared points), coincident, intersecting (they meet at a point) or skew: they neither meet nor are parallel, something that exists only in space
Line and plane	\( d \cdot n \)	If \( \vec{d} \cdot \vec{n} \ne 0 \) they meet at a point; if \( \vec{d} \cdot \vec{n} = 0 \), the line is parallel to the plane, and it is contained in it if, in addition, one of its points satisfies the plane’s equation
Two planes	The normals	Proportional normals: parallel or coincident. Otherwise, they meet in a line

Working out relative positions is, once again, solving a linear system: two planes that intersect are a consistent system with infinitely many solutions and one free parameter —the line of intersection—, and two distinct parallel planes give an inconsistent system. Everything in systems of linear equations applies here under geometric names.

05Distances and angles

\[ d (P, \pi) = \frac{\left| A x_0 + B y_0 + C z_0 + D \right|}{\sqrt{A^2 + B^2 + C^2}} \quad d (P, r) = \frac{\left| \vec{P_0 P} \times \vec{d} \right|}{\left| \vec{d} \right|} \] The first is the projection of the vector going from the plane to the point onto the unit normal. The second comes from the area of the parallelogram divided by the base.

Angle between two planes: the angle between their normals, cos α = |n₁ · n₂| / (|n₁| |n₂|).
Angle between two lines: that of their direction vectors, with the same formula.
Angle between a line and a plane: the complement of the angle formed by the direction vector and the normal: sin α = |d · n| / (|d| |n|).

Lab · plane, point and another plane

The coefficients are those of the normal. The computation is shown worked out, not just the result.

π₁: x + y + z + = 0

P: (, , )

π₂: x + y + z + = 0

Distance from the point to plane π₁

The two planes with respect to each other

06Where this shows up

Drafting and CAD. Every part modeled in three dimensions is made of planes and lines: views, sections and intersections are computed this way. It is the foundation of technical drawing and computer-aided design.
Antenna pointing. The direction of maximum radiation is a vector; the elevation angle and the azimuth are, exactly, the angles between that vector and reference planes.
Positioning. A GPS solves the intersection of spheres; a time-difference location, the intersection of hyperboloids. The setup is the same: analytic geometry with systems.
Working planes. The optimal tilt of a solar panel, the plane of a printed circuit board relative to the enclosure or the alignment of a sensor are computed with angles between normals.

07In the lab

Exercise 1 · From the point to the plane

Find the equation of the plane through three chosen points, verify that all three satisfy it and compute the distance from a fourth point. Check with the lab above.

Exercise 2 · Skew lines

Propose two lines in space and decide their relative position. If they are skew, compute the distance between them with the scalar triple product: d = |(P₂ − P₁) · (d₁ × d₂)| / |d₁ × d₂|.

Exercise 3 · Measuring it in the workshop

Take a real part —a heat sink, an enclosure— and describe two of its faces as planes using three points measured with a caliper. Compute the angle between them and compare it with the direct measurement with a protractor.

08Common mistakes

Confusing the direction vector with the normal. The direction vector goes along the line; the normal goes perpendicular to the plane.
Writing the symmetric equations with a zero in the denominator.
Forgetting the absolute value in the distance formulas, and getting negative distances.
Assuming that two lines that do not meet are parallel: in space they can be skew.
Not normalizing when projecting onto the normal: the distance has the magnitude of n in the denominator.
Giving the angle between planes as that of the normals without taking the absolute value, and getting the supplementary angle.

09Self-assessment

What is the normal of the plane 2x − y + 3z = 7?

(2, −1, 3): the coefficients of x, y and z.

Distance from the origin to the plane 3x + 4y + 5z − 13 = 0.

|−13| / √(9 + 16 + 25) = 13 / √50 ≈ 1.84.

How do you decide whether a line is parallel to a plane?

By checking that d · n = 0. If, in addition, a point of the line satisfies the plane’s equation, the line is contained in the plane.

What are two skew lines?

Two lines in space that do not meet and are not parallel: they lie in different planes. They do not exist in the plane.

Write the line through (1, 2, 3) with direction (0, 1, −2) in parametric form.

x = 1, y = 2 + t, z = 3 − 2t.

What is the angle between the planes x + y = 0 and x − y = 0?

Normals (1, 1, 0) and (1, −1, 0): cos α = |1 − 1| / (√2·√2) = 0, so α = 90°.

10Further reading

Charles H. Lehmann. Analytic Geometry. (Spanish edition, “Geometría analítica”), Limusa. The classic treatment of the line and the plane, with a systematic account of relative positions.
Stanley I. Grossman. Álgebra lineal. 6th ed., McGraw-Hill, 2008 (in Spanish). Approaches the line and the plane from vectors, which is the short route for this course.
Howard Anton. Elementary Linear Algebra. 5th ed. (Spanish edition, “Introducción al álgebra lineal”), Limusa Wiley, 2011. Good figures and exercises on distances and angles in space.

← Previous topic

Vectors in R² and R³

Next topic →

Quadratic forms and conics