Position vectors, distance and normalization

A position vector in ℝ² or ℝ³ is a map r: t ↦ (x(t), y(t)) or r: t ↦ (x(t), y(t), z(t)). The distance from the origin at time t is defined as the Euclidean norm

$$\Vert r(t)\Vert =\surd (x(t{)}^2+y(t{)}^2)\text{(R²)}$$

and analogously in ℝ³. Normalization is the operation

$$\hat{r}=r/\Vert r\Vert$$

which produces a unit vector with the same direction as r. This construction underlies all “unit radial” vector fields.

Important

Normalization changes magnitude only, never direction.

Radial and tangential directions

Given a point (x, y) ≠ (0, 0):

• The radial direction is the direction of the vector (x, y).
• A vector is positive radial if its dot product with (x, y) is positive:

$$F(x,y)\cdot (x,y)>0.$$

This encodes “pointing away from the origin”. Component signs are irrelevant.

A vector is tangential to circles centered at the origin if it is orthogonal to the radial direction:

$$F(x,y)\cdot (x,y)=0.$$

This definition is specific to circles and should not be confused with tangency to arbitrary curves.

Tangent vectors to curves

See Tangent Vectors and the Unit Normal on Graph Surfaces for tangent vectors on surfaces in ${\mathbb{R}}^3$ and Arc-Length Parametrization for parametrized curves and arc length.

Graphs y = f(x)

A graph can be viewed as the parametric curve

$$\gamma (x)=(x,f(x)).$$

Its tangent vector is

$${\gamma }^{\prime }(x)=(1,f^{\prime }(x)).$$

The usual slope f′(x) is the ratio of the y‑component to the x‑component of this vector.

Parametric curves

A general curve is given by

$$\gamma (t)=(x(t),y(t)).$$

The tangent vector at t is defined as

$${\gamma }^{\prime }(t)=(x^{\prime }(t),y^{\prime }(t)),$$

where the derivative is taken componentwise. This definition follows directly from the limit of secant displacement vectors and is not an analogy with slopes; the graph case is a special instance.

![important] “Tangent vector” always means the derivative of the curve with respect to its parameter.

Field lines (integral curves)

Let F(x, y) = (Fₓ(x,y), Fᵧ(x,y)) be a vector field in ℝ².

A field line is a curve γ(t) = (x(t), y(t)) such that, for all t,

$${\gamma }^{\prime }(t)\Vert F(\gamma (t)).$$

Parallelism means proportional components, not orthogonality.

Differential equation for field lines

From

$$(x^{\prime }(t),y^{\prime }(t))=\lambda (t)(F_x,F_{\gamma }),$$

one obtains, assuming Fₓ ≠ 0,

$$\frac{y^{\prime }(t)}{x^{\prime }(t)}=\frac{F_{\gamma }(x,y)}{F_x(x,y)}.$$

Along the curve, y depends on x through the parameter t. A small increment Δt produces

$$\Delta x\approx (dx/dt)\Delta t,\Delta y\approx (dy/dt)\Delta t.$$

Taking the ratio and passing to the limit gives

$$\frac{dy}{dx}=\frac{dy/dt}{dx/dt}.$$

Hence the field line equation

$$\boxed{dy/dx=F_{\gamma }(x,y)/F_x(x,y)}.$$

![warning] Dot products are used for orthogonality problems, not for field lines.

Radial and tangential vector fields

A unit radial field in ℝⁿ (n = 2 or 3) is

$$F(x)=x/\Vert x\Vert ,x\ne 0.$$

It has unit magnitude and satisfies F(x) · x > 0.

A purely tangential field in ℝ² with magnitude equal to distance from the origin is

$$F(x,y)=(-y,x).$$

Its norm is √(x² + y²) and it satisfies F · (x,y) = 0.

Parametric motion, velocity, and acceleration

For a trajectory

$$r(t)=(acos(\omega t),bsin(\omega t)),$$

• Velocity is v(t) = r′(t).
• Acceleration is a(t) = r″(t).

Direct differentiation yields

$$a(t)=-{\omega }^{2}r(t).$$

This equation expresses a restoring acceleration proportional to displacement, not damping. Eliminating t gives

$$x^2/a^2+y^2/b^2=1,$$

the Cartesian equation of an ellipse.

Coulomb field and superposition (unit‑vector form)

For a point charge q at position r₀, the electric field at r ≠ r₀ is

$$E(r)=kq\cdot \hat{u}/R^2,$$

where R = ‖r − r₀‖ and \hat u = (r − r₀)/R.

Writing the unit vector explicitly produces the equivalent vector form

$$E(r)=kq(r-r_0)/\Vert r-r_0{\Vert }^3.$$

The inverse‑square law is preserved; the extra power arises from normalizing the direction vector.

Tip

Always separate magnitude law (1/R²) from direction normalization when checking exponents.

See also

• Position Vectors and Coordinate-Free Geometry --- coordinate-free proofs using position vectors