Linear Algebra
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The Tangent Vector

For a differentiable trajectory γ:I→Rm\boldsymbol{\gamma}: I \to \mathbb{R}^m, the tangent vector at t0t_0 is the derivative γ′(t0)\boldsymbol{\gamma}'(t_0). This is the Jacobian of γ\boldsymbol{\gamma} (a m×1m\times 1 matrix, i.e., a column vector):

γ′(t0)=lim⁡h→0γ(t0+h)−γ(t0)h=[γ1′(t0)⋮γm′(t0)]\boldsymbol{\gamma}'(t_0) = \lim_{h\to 0}\frac{\boldsymbol{\gamma}(t_0+h)-\boldsymbol{\gamma}(t_0)}{h} = \begin{bmatrix}\gamma_1'(t_0)\\ \vdots \\ \gamma_m'(t_0)\end{bmatrix}

Geometrically, γ′(t0)\boldsymbol{\gamma}'(t_0) is the velocity vector at t0t_0: its direction is the tangent direction to the curve at the point γ(t0)\boldsymbol{\gamma}(t_0), and its magnitude ∥γ′(t0)∥\|\boldsymbol{\gamma}'(t_0)\| is the speed (rate of arc length traversal).

Formal View

Definition 13.6 — Tangent Vector / Velocity
For a differentiable trajectory γ:I→Rm\boldsymbol{\gamma}: I \to \mathbb{R}^m, the tangent vector (velocity) at t0∈It_0 \in I is
γ′(t0)=dγdt(t0)=lim⁡h→0γ(t0+h)−γ(t0)h\boldsymbol{\gamma}'(t_0) = \frac{d\boldsymbol{\gamma}}{dt}(t_0) = \lim_{h\to 0}\frac{\boldsymbol{\gamma}(t_0+h)-\boldsymbol{\gamma}(t_0)}{h}

The unit tangent vector is T(t0)=γ′(t0)/∥γ′(t0)∥\mathbf{T}(t_0) = \boldsymbol{\gamma}'(t_0)/\|\boldsymbol{\gamma}'(t_0)\| (defined when γ′(t0)≠0\boldsymbol{\gamma}'(t_0) \neq \mathbf{0}).

Example 13.3 — Tangent Vector to a Circle
For γ(t)=(cos⁡t,sin⁡t)\boldsymbol{\gamma}(t) = (\cos t, \sin t): γ′(t)=(−sin⁡t,cos⁡t)\boldsymbol{\gamma}'(t) = (-\sin t, \cos t). At t=0t=0: position (1,0)(1,0), tangent (0,1)(0,1) — pointing upward, perpendicular to the radius, as expected for a circle.

Why This Matters

The tangent vector is the foundation of differential geometry and the physics of motion.

  • Physics: velocity and acceleration of a particle in 3D space
  • Differential geometry: curvature and torsion of curves
  • Robot path planning: ensuring smooth trajectory derivatives for continuous velocity profiles

Quiz

Question 1

For γ(t)=(t2,t3)\boldsymbol{\gamma}(t) = (t^2, t^3), the tangent vector at t=1t=1 is:

Question 2

The magnitude of the tangent vector ∥γ′(t0)∥\|\boldsymbol{\gamma}'(t_0)\| represents the speed of traversal of the curve.

Common Mistakes

  • Confusing the tangent vector (a vector at a point) with the tangent line (the line through that point in the tangent direction).
  • Forgetting to evaluate the derivative at the specific parameter value.
  • Confusing speed (∥γ′∥\|\boldsymbol{\gamma}'\|, a scalar) with velocity (γ′\boldsymbol{\gamma}', a vector).