Difficulty: Intermediate | Prerequisites: Vector operations, basic integration from Calculus II
Parametric equations let you describe lines and curves as functions of a single variable t, which makes it straightforward to compute distances, speeds, and lengths. The speed at any moment is the magnitude of the derivative of the position vector, and the total distance travelled (arc length) is the integral of that speed. Trigonometric identities are the main tool for simplifying the integrands that appear in arc length problems.
Parametrisation
A way of describing a curve or line segment by expressing each coordinate (x, y, z) as a function of a single parameter t. Think of t as a clock: as it ticks from one value to another, the point traces out the path.
Position vector r(t)
The vector-valued function ⟨x(t), y(t), z(t)⟩ that gives the location of a point on the curve at parameter value t.
Velocity vector r'(t)
The derivative of the position vector with respect to t. It points in the direction the curve is heading at that instant, and its magnitude is the speed.
Speed |r'(t)|
The magnitude of the velocity vector at parameter value t. In simple terms, it tells you how fast the point is moving along the curve at that moment.
Arc length
The total distance travelled along a curve from t = a to t = b, obtained by integrating the speed over that interval. Think of it as laying the curve flat and measuring it with a ruler.
Double-angle identities
Trigonometric identities that express functions of 2x in terms of functions of x (and vice versa). They are the main simplification tool when arc length integrands contain sin²x or cos²x terms.
Given two points P and Q with position vectors p and q, the line segment from P to Q is parametrised by:
r(t) = p + t(q − p), where t ∈ [0, 1]
At t = 0 you are at P; at t = 1 you are at Q; at t = 0.5 you are at the midpoint.
This parametrisation is useful whenever you need to find a specific point along the segment, such as the closest point on a line to a given location.
When finding the shortest distance between two spheres, parametrise the line connecting their centres. The closest points on the surfaces lie along this line, offset by each sphere's radius.
Given r(t) = ⟨x(t), y(t), z(t)⟩, the velocity vector is r'(t) = ⟨x'(t), y'(t), z'(t)⟩.
The speed at time t is the magnitude: |r'(t)| = √((dx/dt)² + (dy/dt)² + (dz/dt)²).
Speed is always non-negative. It equals zero only when the particle is momentarily stationary.
The arc length from t = a to t = b is: L = ∫ₐᵇ |r'(t)| dt.
This integral sums up the infinitesimal distances along the curve.
When the integrand involves sin² or cos² terms, apply trigonometric identities before integrating.
sin²x = (1 − cos 2x) / 2
cos²x = (1 + cos 2x) / 2
sin²x + cos²x = 1
cos 2x = cos²x − sin²x
sin 2x = 2 sin x cos x
These identities convert squared trig functions into expressions involving double angles, which are typically much easier to integrate.
\mathbf{r}(t) = \mathbf{p} + t(\mathbf{q} - \mathbf{p}), \quad t \in [0,1]|\mathbf{r}'(t)| = \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2 + \left(\frac{dz}{dt}\right)^2}L = \int_a^b |\mathbf{r}'(t)|\, dt\sin^2 x = \frac{1 - \cos 2x}{2}, \qquad \cos^2 x = \frac{1 + \cos 2x}{2}Parametric curves model the paths of moving objects: projectiles, satellites, robotic arms. The arc length formula is how GPS systems calculate the distance you have actually travelled along a winding road, as opposed to the straight-line distance between two points. Trig identities show up any time periodic signals (sound waves, alternating current) need to be integrated or simplified.
Students often confuse speed with velocity. Speed is a scalar (always non-negative); velocity is a vector (it has direction). The speed is the magnitude of the velocity.
A common mistake is forgetting to take the magnitude before integrating for arc length. Integrating r'(t) directly gives displacement (a vector), not distance (a scalar).
Students sometimes assume t represents time. It can, but it need not: t is simply a parameter. The interval [0, 1] for a line segment is a choice of parametrisation, not a time interval.
When applying trig identities, students occasionally use the wrong half-angle form. Double-check: sin²x uses (1 − cos 2x)/2, not (1 + cos 2x)/2.
⚠️ Line segment parametrisation is a building block for distance-between-objects problems. Be ready to write down r(t) = p + t(q − p) without hesitation.
⚠️ Arc length computation is a classic midterm question. The typical pattern: differentiate each component, compute the magnitude, simplify with trig identities, then integrate.
⚠️ Know your trig identities by heart. The exam will not provide them, and the arc length integral is often impossible to evaluate without them.
⚠️ Questions sometimes ask for the speed at a particular value of t rather than the full arc length. Read carefully: speed is |r'(t)| evaluated at one point, not an integral.
Fill in the blank: for a line segment from P to Q, r(t) = p + t(q − p). At t = 0.5, you are at the ______.
Midpoint.
True or false: speed can be negative.
False. Speed is a magnitude and is always ≥ 0.
Fill in the blank: arc length is the integral of ______ from t = a to t = b.
Speed, |r'(t)|.
True or false: sin²x = (1 + cos 2x) / 2.
False. sin²x = (1 − cos 2x) / 2.
Fill in the blank: integrating the velocity vector gives ______, not distance.
Displacement.
Q: Parametrise the line segment from P = (1, 2, 3) to Q = (4, 0, −1). What is the position at t = 1/3?
A: r(t) = ⟨1, 2, 3⟩ + t⟨3, −2, −4⟩ = ⟨1 + 3t, 2 − 2t, 3 − 4t⟩. At t = 1/3: r(1/3) = ⟨2, 4/3, 5/3⟩.
Q: A particle has position r(t) = ⟨cos t, sin t, t⟩. Find its speed at any time t.
A: r'(t) = ⟨−sin t, cos t, 1⟩. Speed = √(sin²t + cos²t + 1) = √(1 + 1) = √2. The speed is constant.
Q: Find the arc length of r(t) = ⟨cos t, sin t, t⟩ from t = 0 to t = 2π.
A: Since |r'(t)| = √2 (constant), L = ∫₀²π √2 dt = 2π√2.
Q: Simplify the integrand √(1 + sin²t + cos²t).
A: sin²t + cos²t = 1, so the expression becomes √(1 + 1) = √2.
Q: Find the arc length of r(t) = ⟨t, t², 0⟩ from t = 0 to t = 1.
A: r'(t) = ⟨1, 2t, 0⟩. Speed = √(1 + 4t²). Arc length = ∫₀¹ √(1 + 4t²) dt. This requires a trig substitution (let 2t = tan θ) or a standard integral formula.
parametric curves, parametrisation, line segment, position vector, velocity vector, speed, arc length, arc length integral, trigonometric identities, double-angle formula, half-angle formula, sin squared, cos squared, helix, distance between spheres, MATH 215 Midterm 1, Calculus III, multivariable calculus, University of Michigan