Difficulty: Intermediate | Prerequisites: Parametric equations, dot product, cross product, magnitude
This material connects vectors to motion. A curve in 3D space is described by a vector-valued function r(t), and differentiating it gives you velocity, acceleration, and speed. From there, you can measure the total distance travelled (arc length), how sharply the path turns (curvature), and which part of the acceleration changes speed versus changes direction (tangential and normal components). These ideas form the bridge between static vector geometry and the dynamic, calculus-heavy parts of the course.
Differentiate position to get velocity, differentiate velocity to get acceleration, and take the magnitude of velocity to get speed. Arc length integrates speed. Curvature measures turning sharpness. Acceleration splits into a speed-changing part (tangential) and a direction-changing part (normal).
Position vector r(t)
The vector-valued function r(t) = ⟨x(t), y(t), z(t)⟩ that traces out a curve as t varies. Think of it as: the coordinates of a moving particle at time t.
Velocity v(t)
The derivative of the position vector: v(t) = r'(t). It points in the direction of motion and its magnitude is the speed. In simple terms, velocity is how fast and which way.
Acceleration a(t)
The derivative of velocity: a(t) = v'(t) = r''(t). It describes how the velocity is changing. In simple terms, acceleration captures both speeding up/slowing down and turning.
Speed |v(t)|
The magnitude of the velocity vector: √((x')² + (y')² + (z')²). Speed is a scalar, always non-negative. Think of it as: the length of the velocity vector, not the velocity itself.
Arc length (L)
The total distance travelled along a curve, computed by integrating speed over the parameter interval. In simple terms, it is the odometer reading for the curve.
Curvature (κ)
A measure of how sharply a curve turns at a given point. Computed as |v × a| / |v|³. In simple terms, high curvature means a tight turn; low curvature means the curve is nearly straight.
Tangential acceleration (a_T)
The component of acceleration that changes the speed: a_T = (v · a) / |v|. Think of it as: the gas pedal and the brake.
Normal acceleration (a_N)
The component of acceleration that changes the direction: a_N = |v × a| / |v|. Think of it as: the steering wheel.
The derivative ladder is the chain of relationships that connects position to motion:
Position: r(t) = ⟨x(t), y(t), z(t)⟩
Velocity: v(t) = r'(t) (differentiate position)
Acceleration: a(t) = v'(t) (differentiate velocity)
Speed: |v(t)| = √((x')² + (y')² + (z')²) (take the magnitude of velocity)
Each step is just differentiation or taking a magnitude. The ladder runs in one direction (down), so if you are given position, you differentiate to get everything else.
The arc length L of a curve from t = start to t = end is the integral of speed:
L = ∫ (from start to end) |v(t)| dt = ∫ √((x')² + (y')² + (z')²) dt
Pro-tip: if the expression under the square root is messy, look for a way to factor it into a perfect square (...)² so the root disappears. This simplification appears in exam problems more often than not.
Curvature κ measures how sharply the curve bends:
κ = |v × a| / |v|³
How to solve: if you are given a specific time t, plug that value into v(t) and a(t) first to get simple numerical vectors, then compute the cross product and magnitudes. Working with numbers is far easier than carrying symbolic expressions through the cross product.
Acceleration breaks into two orthogonal parts:
a = a_T T + a_N N
a_T (tangential) = (v · a) / |v| : the part that changes your speed
a_N (normal) = |v × a| / |v| : the part that makes you turn
Notice that a_T uses the dot product and a_N uses the cross product. The tangential component is a scalar projection of acceleration onto velocity; the normal component is what is left over.
\mathbf{v}(t) = \mathbf{r}'(t), \quad \mathbf{a}(t) = \mathbf{v}'(t)|\mathbf{v}(t)| = \sqrt{(x')^2 + (y')^2 + (z')^2}L = \int_{\text{start}}^{\text{end}} |\mathbf{v}(t)|\, dt\kappa = \frac{|\mathbf{v} \times \mathbf{a}|}{|\mathbf{v}|^3}a_T = \frac{\mathbf{v} \cdot \mathbf{a}}{|\mathbf{v}|}, \quad a_N = \frac{|\mathbf{v} \times \mathbf{a}|}{|\mathbf{v}|}Speed and velocity are not the same. Velocity is a vector (direction + magnitude). Speed is the magnitude of velocity (a scalar, always non-negative).
Students sometimes try to find arc length by computing the distance between the start and end points. That gives the straight-line distance, not the distance along the curve.
Curvature κ is always non-negative. If your calculation gives a negative number, you have likely dropped the absolute value on the cross product.
a_T and a_N are scalar components, not vectors. They tell you how much acceleration goes in each direction, not the direction itself.
⚠️ The derivative ladder is foundational. If a problem gives you r(t), you will likely need to find v(t), a(t), and |v(t)| before answering.
⚠️ Arc length integrals often simplify to a perfect square under the root. Look for this before integrating the hard way.
⚠️ For curvature at a specific time, plug in t first, then compute. Students who keep t symbolic through the cross product tend to make algebraic errors.
⚠️ Know which operation goes with which acceleration component: dot product for tangential (a_T), cross product for normal (a_N).
Curvature determines the banking angle of a road on a curve and the turning radius of an aircraft. Components of acceleration matter in roller coaster design: the tangential component is the thrill of speeding up, and the normal component is the G-force you feel in a turn.
Fill in the blank: speed is the ______ of the velocity vector.
Magnitude.
True or false: arc length can be found by subtracting the start point from the end point.
False. Arc length requires integrating speed over the parameter interval.
Fill in the blank: a_T uses the ______ product; a_N uses the ______ product.
Dot; cross.
True or false: curvature can be negative.
False. Curvature is always non-negative.
Q: Given r(t) = ⟨t, t², t³⟩, find v(t) and a(t).
A: v(t) = r'(t) = ⟨1, 2t, 3t²⟩. a(t) = v'(t) = ⟨0, 2, 6t⟩.
Q: Find the speed at t = 1 for r(t) = ⟨t, t², t³⟩.
A: v(1) = ⟨1, 2, 3⟩. |v(1)| = √(1 + 4 + 9) = √14.
Q: Find a_T at t = 1 for r(t) = ⟨t, t², t³⟩.
A: v(1) = ⟨1, 2, 3⟩, a(1) = ⟨0, 2, 6⟩. v · a = 0 + 4 + 18 = 22. |v| = √14. a_T = 22/√14.
The derivative ladder is the starting point for all motion problems in the course. Curvature connects to the TNB frame (tangent, normal, binormal), which some courses cover in depth. Arc length reparametrisation appears in line integrals later in the term.
derivative ladder, position velocity acceleration, vector-valued function, arc length, arc length integral, curvature, kappa, tangential acceleration, normal acceleration, components of acceleration, TNB frame, speed, Calc III, Calculus III, multivariable calculus, parametric curves 3D