Math101 · Independent Ontario learning library
Math101learn.math101.caVectors in Three Dimensions
Three-dimensional vectors extend component methods into space using x-, y-, and z-directions.
Adding a $z$-component lets vectors describe displacement, force, and direction throughout space.
Coordinate space
A point in three-dimensional Cartesian space is $(x,y,z)$. The coordinate axes are mutually perpendicular, and coordinate planes include the $xy$-, $xz$-, and $yz$-planes.
A vector is written
Its components represent change in the $x$, $y$, and $z$ directions.
Magnitude
The magnitude of $\vec v=\langle a,b,c\rangle$ is
This extends the Pythagorean theorem: first combine two perpendicular components, then combine the result with the third.
Vector between two points
From $A(x_1,y_1,z_1)$ to $B(x_2,y_2,z_2)$,
The distance between the points is $\|\overrightarrow{AB}\|$. Reversing the endpoints reverses every component but leaves distance unchanged.
Worked example
Vector operations
Add and subtract corresponding components:
Scalar multiplication multiplies every component. These operations retain the same geometric meanings as in two dimensions.
Unit vectors and direction
For nonzero $\vec v$,
is a unit vector in the same direction. Multiplying a unit direction by a desired magnitude creates a vector with that direction and size.
This is useful for force and velocity vectors given by magnitude plus a line of action.
Direction angles
If $\alpha$, $\beta$, and $\gamma$ are the angles a nonzero vector makes with the positive $x$-, $y$-, and $z$-axes, then
The direction cosines satisfy
Linear combinations and span
An expression $s\vec u+t\vec v$ is a linear combination. In space, two nonparallel vectors based at a common point span a plane, while three suitably independent vectors can span all of three-dimensional space.
This idea underlies parametric equations for lines and planes.
Coplanarity
Vectors are coplanar if they lie in or parallel to one common plane. The scalar triple product
equals zero precisely when three vectors are coplanar, assuming the operations are defined.
Applications
Three-dimensional vectors model aircraft motion, structures, forces, navigation, graphics, and geometry. A coordinate system and units must be stated; otherwise a correct component calculation may have unclear physical meaning.
Resultants are found by combining components, then reporting magnitude and direction.
Common mistakes
Dropping the $z$-component in magnitude. All three squared components contribute.
Using $A-B$ for $\overrightarrow{AB}$. Endpoint minus start point gives the direction.
Dividing by the squared magnitude for a unit vector. Divide by $\|\vec v\|$.
Treating direction angles as planar angles that sum to $90^\circ$. Their cosine squares, not the angles, obey the identity.
Mixing points and vectors. A point is a location; a vector is a displacement or direction.
Quick self-check
- Are all three components included with correct signs?
- Does endpoint subtraction match the requested direction?
- Is magnitude a nonnegative square root?
- Does a unit vector have magnitude $1$?
- Do direction cosines match components divided by magnitude?
- Are the geometric interpretation and units clear?
Related topics
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
From A = (1, −2, 3) to B = (5, 1, −1), what are vector AB and its magnitude?
- AB = ⟨5 − 1, 1 − (−2), −1 − 3⟩ = ⟨4, 3, −4⟩.
- Magnitude = √(4² + 3² + (−4)²) = √41.
End of lesson
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