6.2 A Line and Its Complement

The projection of the last section split a vector into two perpendicular pieces: one in the plane, one along the normal line. That pairing — plane and normal line — is no accident of the example. Every subspace $V$ has an orthogonal complement $V^{⟂}$ : the set of all vectors perpendicular to every vector of $V$ . The complement is itself a subspace (perpendicularity survives adding and scaling), and the two mesh so tightly that each determines the other completely.

The figure makes the pairing mechanical. The red line consists of all multiples of the direction vector $open parenthesis a, b, 1 close parenthesis$ , with $a$ and $b$ on sliders. Which vectors $(x, y, z)$ are perpendicular to it? Exactly those with $(x, y, z) \cdot (a, b, 1) = 0$ , that is, $a x + b y + z = 0$ — the equation of a plane through the origin, drawn in the scene as the surface $z = - a x - b y$ . The scene never computes an angle; the plane is perpendicular to the line by construction, because its equation is the perpendicularity condition itself.

A solid red line through the origin piercing a semi-transparent plane, meeting it at a perfect right angle. Two sliders a and b tilt the line's direction; the plane automatically re-tilts to stay exactly perpendicular to the line, so the pair always looks like a pole through a sheet. — The red line of all multiples of $open parenthesis a, b, 1 close parenthesis$ and its orthogonal complement, the plane $z = - a x - b y$ , whose normal is the line's own direction. The sliders $a$ and $b$ tilt the line; the plane re-tilts in lockstep to stay exactly perpendicular.

Explore the figure

At the defaults $a equals 0.6$ , $b = - 0.4$ , orbit until the plane is edge-on. Line and plane meet like a pole through a sheet — the right angle holds from every viewing direction that shows the plane as a line.
Drag $a$ slowly across its range. The line swings one way; the plane counter-tilts instantly. No slider position breaks the perpendicularity, because the plane's formula is the perpendicularity condition.
Set $a equals 0$ and $b equals 0$ . The line becomes the $z$ -axis and the complement flattens into the $x y$ -plane — the most familiar line–plane pair in the book, revealed as an orthogonal complement pair.
Count dimensions as you drag: one for the line, two for the plane, three in total, at every slider position. The complement absorbs every direction the line does not use.

Every vector splits, uniquely

The complement pairing is what makes projection bookkeeping exact. Given any vector $𝐰$ and the line direction $𝐝 equals open parenthesis a, b, 1 close parenthesis$ , the piece of $𝐰$ along the line is $((𝐰 \cdot 𝐝) / (𝐝 \cdot 𝐝)) 𝐝$ — the projection onto the line — and what remains is perpendicular to $𝐝$ , hence in the plane. So $𝐰$ = line part + plane part, always. Uniqueness comes from the only vector the two subspaces share: a vector in both is perpendicular to itself, so $𝐰 \cdot 𝐰 = 0$ forces it to be the zero vector. Two decompositions would differ by a shared nonzero vector, and there is none to differ by.

Example 1 Split a vertical vector at the default sliders

With $a equals 0.6$ and $b = - 0.4$ , decompose $𝐰 equals open parenthesis 0, 0, 1.52 close parenthesis$ into its line part and its complement part, and verify both pieces land where claimed.

The direction is $𝐝 = (0.6, - 0.4, 1)$ with $𝐝 \cdot 𝐝 = 0.36 + 0.16 + 1 = 1.52$ , and $𝐰 \cdot 𝐝 = 0 + 0 + 1.52 = 1.52$ , so the coefficient is $1.52 / 1.52 equals 1$ : the line part is $𝐝$ itself, $open parenthesis 0.6, minus 0.4, 1 close parenthesis$ . The complement part is the remainder $𝐰 - 𝐝 = (- 0.6, 0.4, 0.52)$ . Check it lies in the plane $z = - 0.6 x + 0.4 y$ : the right side gives $- 0.6 (- 0.6) + 0.4 (0.4) = 0.36 + 0.16 = 0.52$ , matching. Check perpendicularity to the line: $(- 0.6) (0.6) + (0.4) (- 0.4) + (0.52) (1) = - 0.36 - 0.16 + 0.52 = 0$ . A vertical vector, no component of which looks special, splits cleanly into one piece along the tilted line and one in its tilted complement.

An original work of XYZ Homework, built around interactive XYZ 3D figures. Its chapter sequence is aligned to Interactive Linear Algebra (Margalit & Rabinoff, Georgia Tech, GNU FDL); this work is original, copies nothing from it, and is not affiliated with or endorsed by its authors. License: CC-BY-NC-SA-4.0.