MATH 126, Section 12.5 | Calculus III | Stewart 9th ed.
Difficulty: Intermediate | Prerequisites: Section 12.1–12.4 (vectors, dot product, cross product)
Section 12.5 asks you to reason about how lines and planes relate in three-dimensional space. Many 2D intuitions break down here because lines in R³ can be skew (neither parallel nor intersecting). Planes are better behaved: two distinct planes either intersect in a line or are parallel, full stop. Knowing which geometric relationships force parallelism and which do not is the conceptual backbone of this section.
Normal vector
A vector perpendicular to a plane. For the plane ax + by + cz = d, the normal vector is n = ⟨a, b, c⟩. Every plane has a unique direction of normal (up to scalar multiples).
In simple terms, it is the arrow that sticks straight out of a flat surface.
Direction vector
A vector parallel to a line. For parametric equations x = x₀ + at, y = y₀ + bt, z = z₀ + ct, the direction vector is v = ⟨a, b, c⟩.
Think of it as the arrow pointing along the track a line follows.
Parallel lines
Two lines whose direction vectors are scalar multiples of each other. They never intersect and maintain constant separation.
Skew lines
Lines in R³ that are not parallel and do not intersect. They lie in different planes entirely. This concept does not exist in 2D.
Think of it as two lines that pass each other in space without ever touching, like a highway overpass and the road beneath it.
Parallel planes
Two planes whose normal vectors are scalar multiples of each other. They never intersect.
Perpendicular (to a plane)
A line is perpendicular to a plane when the line's direction vector is parallel to the plane's normal vector. Two planes are perpendicular when their normals are perpendicular (dot product equals zero).
These results all concern R³ specifically. In 2D, several of them would have different answers.
Two lines parallel to a third line are parallel. (TRUE) Direction vectors of the first two are each scalar multiples of the third's, so they are scalar multiples of each other.
Two lines perpendicular to a third line are not necessarily parallel. (FALSE) In R³, there are infinitely many directions perpendicular to a given line. Picture the x-axis: both the y-axis and z-axis are perpendicular to it, yet they are not parallel to each other.
Two planes parallel to a third plane are parallel. (TRUE) Their normals are each parallel to the third plane's normal, so they are parallel to each other.
Two planes perpendicular to a third plane are not necessarily parallel. (FALSE) The xz-plane and yz-plane are both perpendicular to the xy-plane, yet they intersect along the z-axis.
Two lines parallel to a plane are not necessarily parallel. (FALSE) A plane contains infinitely many directions. Two lines can each be parallel to the same plane while pointing in completely different directions.
Two lines perpendicular to a plane are parallel. (TRUE) Each line's direction vector must be parallel to the plane's normal, and there is only one normal direction. So both direction vectors are parallel.
Two planes parallel to a line are not necessarily parallel. (FALSE) Many planes can contain a direction parallel to a given line while tilting at different angles. Think of holding a pencil and rotating a sheet of paper around it.
Two planes perpendicular to a line are parallel. (TRUE) Each plane's normal must be parallel to the line's direction vector. Since both normals point the same way, the planes are parallel.
Two planes either intersect or are parallel. (TRUE) Two distinct planes in R³ have exactly three options: they are the same plane, they are parallel (no intersection), or they intersect in a line. Since "the same plane" is sometimes treated as a special case of parallel, the statement holds.
Two lines either intersect or are parallel. (FALSE) In R³, lines can also be skew. This is the classic trap.
A plane and a line either intersect or are parallel. (TRUE) If the line is not parallel to the plane, it must cross it at exactly one point. There is no "skew" option for a line and a plane. The line either sits inside the plane (intersects everywhere), crosses it at one point, or runs parallel without touching it.
"Perpendicular to the same thing means parallel to each other." This is only true when the shared reference is a plane (lines perp to a plane) or a line (planes perp to a line). When two lines are both perpendicular to a third line, or two planes are both perpendicular to a third plane, parallelism is not guaranteed.
"Non-parallel lines must intersect." This holds in 2D but fails in 3D. Skew lines are the counterexample. Always check whether lines actually share a point before concluding they intersect.
"Parallel to a plane means parallel to each other." A plane has infinitely many directions running along its surface. Two lines can both be parallel to the xy-plane while pointing in completely different horizontal directions.
"Lines and planes can be skew." They cannot. A line and a plane either intersect (at a point or along the whole line) or are parallel. The term "skew" applies only to pairs of lines.
True or false: two lines perpendicular to the same plane must be parallel. (True)
True or false: two planes perpendicular to the same plane must be parallel. (False)
Fill in the blank: lines in R³ that are neither parallel nor intersecting are called ____. (Skew)
True or false: a line and a plane can be skew. (False, they either intersect or are parallel)
True or false: two lines parallel to the same plane must be parallel to each other. (False)
Q: The x-axis and y-axis are both perpendicular to the z-axis. Are the x-axis and y-axis parallel?
A: No. They are perpendicular to each other. This is a concrete counterexample showing that two lines perpendicular to the same line need not be parallel.
Q: Give an example of two planes that are both perpendicular to a third plane but are not parallel to each other.
A: The xz-plane and the yz-plane are both perpendicular to the xy-plane, yet they intersect along the z-axis.
Q: Can two lines in R³ fail to be parallel, fail to intersect, and still both exist? What is the term for this?
A: Yes. They are called skew lines. Example: the line along the x-axis and the line x = 0, y = 1, z = t.
Q: A line has direction vector v = ⟨1, 0, 0⟩. A plane has normal vector n = ⟨0, 1, 0⟩. Is the line perpendicular to the plane, parallel to the plane, or neither?
A: v · n = 0, so the line is parallel to the plane (or lies in it). The line is perpendicular to the plane only when v is parallel to n.
Q: Two planes have normal vectors n₁ = ⟨2, 4, −6⟩ and n₂ = ⟨−1, −2, 3⟩. Are the planes parallel?
A: Yes. n₁ = −2 · n₂, so the normals are scalar multiples, making the planes parallel.
This material builds directly on the dot product (Section 12.3) and cross product (Section 12.4). The dot product tells you whether two vectors are perpendicular (dot product = 0) or parallel. The cross product gives you a normal vector from two direction vectors lying in a plane, which is exactly how you find a plane's equation from two lines or from a line and a point.
These relationships reappear in Section 13.1–13.2 when you work with tangent lines and normal planes to space curves, and later in Chapter 14 when gradients produce normal vectors to level surfaces.
lines and planes in 3D, parallel lines R3, skew lines, perpendicular to a plane, normal vector, direction vector, parametric equations of a line, equation of a plane, line-plane intersection, two planes parallel, two lines parallel, Stewart 12.5, Calculus III, Math 126 UW, SCalcET9 12.5.001