Difficulty: Introductory | Prerequisites: Calculus I and II (familiarity with 2D vectors helpful but not required)
Calculus III extends everything you learnt about functions and derivatives into three dimensions. This first lesson lays the groundwork: vectors in 3D space, which you will use constantly for the rest of the course. If you can plot points and draw arrows in the xy-plane, you already have the intuition; you are simply adding a z-axis. Everything here feeds directly into dot products, cross products, planes, curves and surfaces later in the term.
Vectors in 3D are written as ordered triples ⟨a, b, c⟩ and drawn as arrows from the origin along the x-, y- and z-axes. A vector lying along one axis has two zero components; a vector in a coordinate plane has one zero component. To find the component form of a vector between two points, subtract the coordinates of the tail from the tip.
Vector (3D)
A quantity with both magnitude and direction, represented in three-dimensional space as an ordered triple ⟨a, b, c⟩. Think of it as an arrow that tells you how far to go in each of the x, y and z directions.
Component form
The representation of a vector as ⟨x₂ − x₁, y₂ − y₁, z₂ − z₁⟩, computed by subtracting the coordinates of the initial point from the terminal point. In simple terms, it is the "recipe" for getting from one point to another.
Coordinate axes
The three mutually perpendicular lines (x, y, z) that define 3D space. Any vector with two zero components lies entirely along one of these axes.
Coordinate planes
The three flat surfaces formed by pairs of axes: the xy-plane (z = 0), the xz-plane (y = 0) and the yz-plane (x = 0). A vector with exactly one zero component lives in one of these planes.
Octant
One of the eight regions that the three coordinate planes divide 3D space into. The first octant is where x, y and z are all positive, similar to the first quadrant in 2D but with a third dimension.
Position vector
A vector drawn from the origin to a given point. It has the same components as the point's coordinates.
Equivalent vectors
Two vectors that have the same magnitude and direction, regardless of where they are positioned in space. Moving a vector without rotating or scaling it produces an equivalent vector.
A vector like ⟨−3, 0, 0⟩ has non-zero x-component only, so it lies entirely along the x-axis.
⟨0, 5, 0⟩ lies along the y-axis.
Rule: if exactly two components are zero, the vector sits on an axis. Identify which component is non-zero to name the axis.
⟨0, −2, 4⟩ has x = 0, so the vector lies in the yz-plane.
⟨−4, −2, 0⟩ has z = 0, so the vector lies in the xy-plane.
Rule: if exactly one component is zero, the vector lives in the plane formed by the other two axes. The zero component tells you which axis is missing, which names the plane.
When all three components are non-zero, the vector points into one of the eight octants.
⟨5, 5, −3⟩ has positive x, positive y, negative z. That combination places it in octant V (below the xy-plane, in front and to the right).
⟨−3, −4, 2⟩ has negative x, negative y, positive z, placing it in octant III (above the xy-plane, behind and to the left).
Octant numbering follows the quadrant pattern in the xy-plane, then mirrors below for negative z.
Given initial point R and terminal point P, the vector RP is found by subtracting R's coordinates from P's.
Example: P = (0, 4, 0), R = (−3, −2, 0). Then RP = ⟨0 − (−3), 4 − (−2), 0 − 0⟩ = ⟨3, 6, 0⟩.
The component form of a vector and the vector between two points are parallel and have the same length. They are effectively the same arrow, just potentially placed at different starting points.
Component form of a vector between two points:
Given P = (x₁, y₁, z₁) and Q = (x₂, y₂, z₂):
\vec{PQ} = \langle x_2 - x_1,\; y_2 - y_1,\; z_2 - z_1 \rangleIdentifying axis, plane or octant:
Zero components | Where the vector lives | Example |
|---|---|---|
Two zeros | Along an axis | ⟨0, 5, 0⟩ lies on the y-axis |
One zero | In a coordinate plane | ⟨0, −2, 4⟩ lies in the yz-plane |
No zeros | In an octant | ⟨5, 5, −3⟩ lies in octant V |
3D vectors model anything with magnitude and direction in physical space: forces on a bridge, wind velocity, the trajectory of a spacecraft. Engineers and physicists decompose these into components along each axis to simplify calculations, which is exactly the skill this lesson builds.
Students often confuse which component is zero with which plane the vector lives in. A zero x-component means the vector is in the yz-plane, not the x-plane (there is no "x-plane").
Octant numbering is not arbitrary. It mirrors quadrant numbering in the xy-plane. The first four octants sit above the xy-plane (positive z), and octants V through VIII mirror them below (negative z).
When computing the component form of a vector from R to P, students sometimes subtract in the wrong order. Always subtract the initial point from the terminal point: P minus R, not R minus P.
Two vectors can be equivalent (same components) even if they start at different points. Position does not affect the vector's identity, only its components do.
⚠️ Expect questions asking you to identify the axis, plane or octant for a given vector. Count the zeros.
⚠️ Computing the component form of a vector from two points is a near-certain exam question. Know the subtraction order cold.
⚠️ Drawing vectors in 3D on paper (isometric-style axes) is tested. Practise sketching on the standard right-hand coordinate system with x coming toward you, y to the right and z upward.
True or false: the vector ⟨0, 0, −7⟩ lies along the z-axis. (True)
Fill in the blank: a vector with z = 0 lies in the ____-plane. (xy)
True or false: ⟨2, −1, 3⟩ lies in the first octant. (False, because y is negative)
Given A = (1, 0, 3) and B = (4, 2, 1), the component form of AB is ⟨____, ____, ____⟩. (⟨3, 2, −2⟩)
True or false: two vectors with the same components but different starting points are equivalent. (True)
Q: A vector has components ⟨0, −3, 5⟩. In which coordinate plane does it lie?
A: The yz-plane, because the x-component is zero.
Q: Given P = (2, 1, −4) and Q = (−1, 5, 0), find the component form of PQ.
A: PQ = ⟨−1 − 2, 5 − 1, 0 − (−4)⟩ = ⟨−3, 4, 4⟩.
Q: The vector ⟨4, −2, 6⟩ lies in which octant?
A: x is positive, y is negative, z is positive, so octant IV (the region with +x, −y, +z).
Q: Points R = (−3, −2, 0) and P = (0, 4, 0). A student writes RP = ⟨−3, −6, 0⟩. What did they do wrong?
A: They subtracted in the wrong order (R minus P instead of P minus R). The correct component form is ⟨3, 6, 0⟩.
Q: Explain why a vector and its position-form equivalent are said to be "the same vector" even though they start at different points.
A: Vectors are defined by magnitude and direction only. Translating a vector to start at the origin does not change either property, so the two are equivalent.
This material connects directly to the dot product and cross product (Lessons 2 and 3), which both depend on writing vectors in component form. It also sets up parametric equations of lines in 3D, where a direction vector plus a point defines the line. Later in the course, surface normals and gradient vectors rely on the same 3D coordinate system and component logic introduced here.
3D vectors, vector components, coordinate plane, coordinate axes, octant, first octant, position vector, equivalent vectors, graphing vectors in three dimensions, component form from two points, yz-plane, xz-plane, xy-plane, right-hand coordinate system, Calculus III vectors, MATH 293, multivariable calculus vectors