Vectors in Three Dimensions

Three-dimensional vectors extend component methods into space using x-, y-, and z-directions.

Three coordinates describe independent movement along three perpendicular axes. A point in space is connected to the coordinate planes by dashed projections. Component arrows build its position one axis at a time.

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

\vec v=\langle a,b,c\rangle=a\mathbf i+b\mathbf j+c\mathbf k.

Its components represent change in the $x$, $y$, and $z$ directions.

Magnitude

The magnitude of $\vec v=\langle a,b,c\rangle$ is

\|\vec v\|=\sqrt{a^2+b^2+c^2}.

This extends the Pythagorean theorem: first combine two perpendicular components, then combine the result with the third.

Worked example

Unit vectors and direction

For nonzero $\vec v$,

\hat v=\frac{\vec v}{\|\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.

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?