Three-Dimensional Coordinates, Vectors, and the Dot Product

A point in a plane is fixed by two numbers; a point in space needs three. The extra coordinate calls for tools to measure distance, name directions, and multiply geometric quantities. A coordinate system — an origin and three mutually perpendicular axes — names points and measures lengths. A vector records a displacement or a force as one object carrying both size and direction. The dot product, the first of two ways to multiply vectors, recovers the angle between them.

Three-dimensional coordinates

Choose a fixed point $O$, the origin, and three directed lines through it that are mutually perpendicular: the $x$-, $y$-, and $z$-axes. The orientation follows the right-hand rule: curl the fingers of your right hand from the positive $x$-axis toward the positive $y$-axis, and your thumb points along the positive $z$-axis. The three axes taken in pairs determine the three coordinate planes — the $xy$-, $yz$-, and $xz$-planes — which cut space into eight octants. The first octant is the one where all three coordinates are positive.

To locate $(a,b,c)$, start at $O$, move $a$ units along the $x$-axis, then $b$ units parallel to the $y$-axis, then $c$ units parallel to the $z$-axis. Dropping a perpendicular from $P$ to the $xy$-plane gives the projection $(a,b,0)$; the point together with its projections spans a rectangular box whose diagonal runs from $O$ to $P$.

Distance and spheres

The Pythagorean theorem, applied twice to that box, extends the plane distance formula to space with no change in form: add one more squared difference.

For example, the distance from $P(2,-1,7)$ to $Q(1,-3,5)$ is

A sphere is the set of points at a fixed distance $r$ (the radius) from a fixed center $C(h,k,l)$. Squaring the distance condition $\left|PC\right|=r$ gives the standard equation directly.

A second-degree equation in $x,y,z$ with equal, positive coefficients on the squared terms is a sphere; completing the square puts it in standard form. Take

Group and complete each square:

so $(x+2{)}^2+(y-3{)}^2+(z+1{)}^2=8$, a sphere with center $(-2,3,-1)$ and radius $2\sqrt{2}$.

Vectors

A vector records a quantity that has both magnitude and direction — displacement, velocity, force. Geometrically it is an arrow: a directed segment whose length is the magnitude and whose heading is the direction. An arrow from $A$ to $B$ is written $\overrightarrow{AB}$. Two arrows with the same length and direction are the same vector, regardless of where they are drawn; only length and direction matter.

Combining vectors

Two operations generate all of vector algebra.

• Addition (the Triangle Law): to form $\vec{u}+\vec{v}$, place the tail of $\vec{v}$ at the tip of $\vec{u}$; the sum runs from the tail of $\vec{u}$ to the tip of $\vec{v}$. Drawing both vectors from a common tail and completing the parallelogram gives the same sum along the diagonal (the Parallelogram Law), which shows addition is commutative.
• Scalar multiplication: for a real number $c$ (a scalar), the vector $c\vec{v}$ has length $\left|c\right|$ times that of $\vec{v}$, with the same direction if $c>0$ and the opposite direction if $c<0$. The difference is $\vec{u}-\vec{v}=\vec{u}+(-\vec{v})$.

Components

Place a vector's tail at the origin; its tip lands at a point, and those coordinates are the vector's components. This turns geometry into arithmetic.

The vector from $A(2,-3,4)$ to $B(-2,1,1)$ is $⟨-2-2,1+3,1-4⟩=⟨-4,4,-3⟩$. Its magnitude is the length of any representing segment, given by the distance formula:

Arithmetic on vectors is arithmetic on components — add, subtract, and scale componentwise:

These operations satisfy the same eight laws as ordinary addition and scaling.

Standard basis and unit vectors

Three special vectors of length $1$ point along the positive axes:

Any vector decomposes along them, giving an equivalent notation: $⟨a_1,a_2,a_3⟩=a_1\hat{ı}+a_2\hat{ȷ}+a_3\hat{k}$. For instance $⟨1,-2,6⟩=\hat{ı}-2\hat{ȷ}+6\hat{k}$.

A unit vector has length $1$. Any nonzero $\vec{a}$ has a unique unit vector in its direction, found by dividing out the length:

The vector $2\hat{ı}-\hat{ȷ}-2\hat{k}$ has length $\sqrt{4+1+4}=3$, so its direction is captured by the unit vector $\frac{1}{3}(2\hat{ı}-\hat{ȷ}-2\hat{k})=\frac{2}{3}\hat{ı}-\frac{1}{3}\hat{ȷ}-\frac{2}{3}\hat{k}$. Because a force carries a magnitude and a direction, it is a vector too, and the net effect of several forces on a body is their vector sum.

The dot product

Beyond addition and scaling, vectors admit two products. The first, the dot product, returns a scalar that encodes the angle between the two vectors.

For example $⟨2,4⟩\cdot ⟨3,-1⟩=6-4=2$, and $(\hat{ı}+2\hat{ȷ}-3\hat{k})\cdot (2\hat{ȷ}-\hat{k})=0+4+3=7$. The dot product is commutative and distributive, and $\vec{a}\cdot \vec{a}=\Vert \vec{a}{\Vert }^2$.

The geometric meaning

The definition is algebraic, but the dot product has a purely geometric reading in terms of the angle $\theta$ between the two vectors (measured with both tails at a common point, $0\le \theta \le \pi$).

The proof applies the Law of Cosines to the triangle with sides $\vec{a}$, $\vec{b}$, and $\vec{a}-\vec{b}$:

Expanding the left side with $\Vert \vec{a}-\vec{b}{\Vert }^2=(\vec{a}-\vec{b})\cdot (\vec{a}-\vec{b})=\Vert \vec{a}{\Vert }^2-2\vec{a}\cdot \vec{b}+\Vert \vec{b}{\Vert }^2$ and cancelling leaves $\vec{a}\cdot \vec{b}=\Vert \vec{a}\Vert \Vert \vec{b}\Vert \cos \theta$. Solving for the cosine gives the angle directly.

Orthogonality and sign

Since $\Vert \vec{a}\Vert \Vert \vec{b}\Vert >0$ for nonzero vectors, the sign of $\vec{a}\cdot \vec{b}$ is the sign of $\cos \theta$: positive when the vectors point in generally the same direction ($\theta <\pi /2$), zero when they are perpendicular, negative when they point generally apart ($\theta >\pi /2$). The zero case is the most useful.

The vectors $2\hat{ı}+2\hat{ȷ}-\hat{k}$ and $5\hat{ı}-4\hat{ȷ}+2\hat{k}$ are perpendicular because their dot product is $10-8-2=0$.

The angles $\alpha ,\beta ,\gamma$ that $\vec{a}$ makes with the positive axes are its direction angles, and their cosines — the direction cosines — are the components of the unit vector $\vec{a}/\Vert \vec{a}\Vert$: applying the corollary with $\vec{b}=\hat{ı},\hat{ȷ},\hat{k}$ gives $\cos \alpha =a_1/\Vert \vec{a}\Vert$, and likewise for the others.

Projections

How much of $\vec{b}$ lies along the direction of $\vec{a}$? Drop a perpendicular from the tip of $\vec{b}$ onto the line through $\vec{a}$; the shadow it casts is the vector projection ${\mathrm{proj}}_{\vec{a}}\vec{b}$, and its signed length is the scalar projection ${\mathrm{comp}}_{\vec{a}}\vec{b}$.

Since the scalar projection is $\Vert \vec{b}\Vert \cos \theta$ and $\vec{a}\cdot \vec{b}=\Vert \vec{a}\Vert \Vert \vec{b}\Vert \cos \theta$, dividing by $\Vert \vec{a}\Vert$ isolates it. The vector projection is that scalar times the unit vector of $\vec{a}$.

Work as a dot product

Projection defines physical work. When a constant force $\vec{F}$ moves an object along a displacement $\vec{D}$, only the component of $\vec{F}$ along $\vec{D}$ does work. That component is a scalar projection, so the work is the dot product