Lines and Planes in Space

A Line Needs a Direction, Not a Slope

Back in the coordinate plane, a line was pinned down by a slope and a y-intercept — two numbers, and you were done. Space refuses that shortcut. There is no single "slope" in three dimensions, because a line can tilt up-and-down, left-and-right, and forward-and-back all at once. Instead we describe a line the way you would give a walking instruction: stand at a known point, then keep stepping in a fixed direction. The known point is a position vector P_0 = (x_0, y_0, z_0), and the fixed direction is just a vector d = (a, b, c) — the direction vector.

Put those two pieces together and you get the vector equation of a line: r = P_0 + t d, where t is a real number you are free to dial. When t = 0 you sit exactly at P_0; as t grows you slide forward along d; as t goes negative you slide backward. Letting t run over all real numbers sweeps out every point on the line, and not one point more. The same equation, read coordinate by coordinate, gives the parametric form x = x_0 + a t, y = y_0 + b t, z = z_0 + c t.

Two Lines in Space: Meet, Miss, or Run Parallel

In the plane, two distinct lines either cross at one point or stay parallel forever — there is no third option. Space adds a genuinely new one. Two lines in three dimensions can be parallel (same direction), they can intersect (cross at a shared point), or they can be skew: not parallel and yet never touching, like a highway overpass crossing the road below without meeting it. Skew lines simply cannot happen in two dimensions, which is one of the first signs that space is a richer place than the page.

Telling them apart is mechanical once you trust your vector tools. First compare the two direction vectors: if one is a scalar multiple of the other, the lines are parallel (and equal if they share any point). If the directions are not parallel, the lines are either intersecting or skew — set the two parametric forms equal and try to solve for the two parameters. If a consistent solution exists, that is the crossing point; if the equations contradict each other, the lines are skew. The angle between two lines that do meet is read off their directions with the dot product: cos theta = (d_1 . d_2) / (|d_1| |d_2|).

Line 1:  r = (1, 0, 2) + t (1, 1, 0)
Line 2:  r = (0, 1, 5) + s (2, 0, 1)
directions (1,1,0), (2,0,1) not parallel
x: 1 + t = 2s    y: t = 1    z: 2 = 5 + s
z gives s = -3, then x: 1 + t = -6 -> t = -7
but y demands t = 1  -> contradiction => SKEW

A Plane Is Defined by What It Faces

A line was anchored by a direction to travel along. A flat plane flips that idea inside out: it is anchored by the one direction it does not contain — the direction it faces. That perpendicular direction is the normal vector n = (A, B, C). Stand at a known point P_0 on the plane; any other point P is on the plane exactly when the little arrow from P_0 to P lies flat in the plane, which means it is perpendicular to n. And "perpendicular" is precisely what the dot product detects: n . (P - P_0) = 0. That single equation is the plane.

Multiply that dot product out and the famous scalar form drops out: A x + B y + C z = D, where D = n . P_0 collects the constant. The coefficients in front of x, y, z are not arbitrary decoration — they are the components of the normal vector, read straight off the equation. So the plane 2x - y + 3z = 7 faces the direction (2, -1, 3), and you knew its orientation before doing a single calculation. This is one of the quiet gifts of the vector viewpoint: the equation wears its geometry on its sleeve.

Distances and Angles, Done with Dots and Crosses

Here is the payoff. Suppose you want the distance from a point Q out in space to a plane with normal n through point P_0. Draw the arrow P_0Q; only its component along n actually reaches toward the plane, and the projection onto a unit normal measures exactly that. The result is the clean formula for the distance from a point to a plane: take |n . (Q - P_0)| and divide by |n|. No dropping of perpendicular feet, no solving systems — one dot product and one length.

1. Identify the plane's normal n = (A, B, C) by reading the coefficients of x, y, z straight off the equation A x + B y + C z = D.
2. Pick any point P_0 actually on the plane, or simply use the constant: the signed quantity is (A x_Q + B y_Q + C z_Q - D).
3. Take the absolute value of that number to discard the sign — distance is never negative.
4. Divide by |n| = sqrt(A^2 + B^2 + C^2), the length of the normal, to get the true perpendicular distance.

Angles between planes work the same way through their normals. The dihedral angle — the angle along the crease where two planes meet, like the opening angle of a half-shut book — is found from cos theta = (n_1 . n_2) / (|n_1| |n_2|). Two planes are parallel when their normals are parallel, and perpendicular when their normals satisfy n_1 . n_2 = 0. Be a little careful: the dot product can hand you the supplement of the angle you wanted, so take the acute value if it is the sharp opening you are after.

Where Lines Meet Planes — and What Comes Next

Lines and planes meet in a satisfying way. Substitute the line's parametric form x = x_0 + a t, y = y_0 + b t, z = z_0 + c t straight into the plane's equation A x + B y + C z = D. Everything collapses to a single linear equation in the lone unknown t. Solve it, feed that t back into the line, and you have the piercing point. If the line is parallel to the plane — meaning its direction d is perpendicular to the normal, d . n = 0 — the t-equation has no solution (the line misses) or is always true (the line lies inside the plane). One dot product, d . n, tells you which case you are in before you solve anything.

Step back and notice the pattern that has carried this whole guide. A line is one point plus a direction to follow; a plane is one point plus a direction to face. The dot product measures the facing, the cross product builds a perpendicular when you need one, and lengths come from the norm. Every distance, every angle, every intersection in this section was just those three operations from earlier in the rung, used in space. Nothing new had to be invented — only pointed in a third direction.

One honest caveat before the next guide. We have treated space as flat, three-dimensional, and ruled by ordinary right-angle geometry — the comfortable world where the Pythagorean theorem holds and parallel really means never-meeting. That assumption is exactly right here and will carry you a long way. Much later in this ladder, when surfaces start to curve and the geometry becomes intrinsic, even the meaning of a "straight line" gets renegotiated. For now, the last guide in this rung returns to the cross product to measure not area but volume — three vectors, one box, and a single number that ties them together.