MISSION · Geometry
Vectors in Three Dimensions
One more coordinate, almost no new rules, and one genuinely new operation
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The idea in one sentence
Adding a third axis changes almost nothing about how vectors work — and unlocks one operation that has no two-dimensional counterpart.
What carries over unchanged
A point now needs three numbers, \((x, y, z)\), with the \(z\)-axis perpendicular to the screen. The convention is right-handed: point the fingers of your right hand along \(x\), curl them towards \(y\), and your thumb points along \(z\).
Every rule from two dimensions survives by gaining a third slot.
Addition and scaling, component by component: \((1,2,3) + (4,0,-1) = (5,2,2)\).
Magnitude, by Pythagoras applied twice:
\[|\mathbf{v}| = \sqrt{x^2 + y^2 + z^2}\]So \(|(2,3,6)| = \sqrt{4+9+36} = \sqrt{49} = 7\). The double application is worth seeing once: \(\sqrt{x^2+y^2}\) is the diagonal across the floor, and that diagonal with the height \(z\) forms a second right triangle standing upright.
The dot product, with one more term:
\[\mathbf{a}\cdot\mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3\]and it still gives \(|\mathbf{a}||\mathbf{b}|\cos\theta\), still returns zero exactly when the two are perpendicular. \((1,0,0)\cdot(0,1,0) = 0\), as two axes should.
Vectors are often written in the basis \(\mathbf{i}, \mathbf{j}, \mathbf{k}\) — the unit vectors along the three axes — so \((2,3,6)\) becomes \(2\mathbf{i} + 3\mathbf{j} + 6\mathbf{k}\). Same object, different notation.
The cross product, which is genuinely new
In a plane, “perpendicular to this vector” has one answer up to sign. In space, a single vector has a whole plane of perpendiculars — but two vectors pin down a direction perpendicular to both.
\[\mathbf{a}\times\mathbf{b} = (a_2b_3 - a_3b_2,\; a_3b_1 - a_1b_3,\; a_1b_2 - a_2b_1)\]Unlike the dot product, this returns a vector. Two properties define what it is for:
Direction: perpendicular to both inputs, with the sense given by the right-hand rule — fingers along \(\mathbf{a}\), curl towards \(\mathbf{b}\), thumb gives \(\mathbf{a}\times\mathbf{b}\).
Magnitude: \(|\mathbf{a}\times\mathbf{b}| = |\mathbf{a}||\mathbf{b}|\sin\theta\), which is the area of the parallelogram the two vectors span.
Check it on the axes: \(\mathbf{i}\times\mathbf{j} = \mathbf{k}\). Two perpendicular unit vectors give a unit vector perpendicular to both, and the parallelogram they span is a unit square.
Order matters, and that is not a defect
\[\mathbf{a}\times\mathbf{b} = -(\mathbf{b}\times\mathbf{a})\]Swapping the inputs reverses the output. The right-hand rule run the other way points the opposite way, so the sign flip is the geometry, not a quirk of the formula.
Compare the dot product, which is symmetric — \(\cos\) does not care about order, and \(\sin\) does.
It also follows that \(\mathbf{a}\times\mathbf{a} = \mathbf{0}\), and more generally that parallel vectors have zero cross product, since \(\sin 0 = 0\) and they span no area at all. For nonzero vectors, dot product zero means perpendicular; cross product zero means parallel. The two operations test opposite things.
What it is used for
- Normal vectors. Two vectors lying in a plane cross to give a vector perpendicular to that plane, which is exactly what the plane’s equation needs. This is the main use in analytic geometry, and the main use in computer graphics for lighting a surface.
- Torque and angular momentum. \(\boldsymbol\tau = \mathbf{r}\times\mathbf{F}\). Push a door at the hinge and the torque is zero, because \(\mathbf{r}\) and \(\mathbf{F}\) are parallel — which matches everyone’s experience of doors.
- Areas and volumes. The parallelogram area comes free, and \(\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})\) gives the volume of the box on three vectors.
- The magnetic force on a moving charge, \(q\mathbf{v}\times\mathbf{B}\), whose sideways direction is the reason electric motors turn.
Common mix-ups
- Returning a scalar from a cross product, or a vector from a dot product. Dot gives a number; cross gives a vector.
- Assuming \(\mathbf{a}\times\mathbf{b} = \mathbf{b}\times\mathbf{a}\). It is the negative.
- Sign errors in the middle component. That term is \(a_3b_1 - a_1b_3\), which runs the opposite way to the other two — a common source of quiet errors.
- Using the cross product in two dimensions. It only exists in three. A 2D problem can be embedded by setting \(z = 0\), and the result points straight out of the plane.
- Confusing “perpendicular” with “zero”. For nonzero vectors, zero dot product means perpendicular; zero cross product means parallel. The zero vector has no direction, so these direction words do not apply to it.
Check yourself
Find \(|(2,3,6)|\), evaluate \((1,0,0)\cdot(0,1,0)\), and say what kind of object a cross product is.
\(\sqrt{4+9+36} = 7\). The dot product of two perpendicular axes is 0. And the cross product returns a vector, perpendicular to both of its inputs.
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