← Calculus III (Multivariable)

Unit 2: Vector-Valued Functions

A moving point's position becomes one vector-valued function, and its derivatives hand you velocity, curvature, torsion, and the frame riding the curve.

19 topics · 69 questions

Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.

3.1 Vector-Valued Functions and Parametrizing a Curve in Space

  1. How it worksHow do three coordinate functions of t combine into a single vector-valued function tracing a curve?
  2. Worked exampleFind a vector-valued function for the line segment from (1, 0, 2) to (3, 4, -1).
  3. Common mistakeWhy isn't a vector-valued function's graph the same kind of picture as an ordinary function's graph?
Your turn1 of 2
A line segment is parametrized as r(t) = P + t(Q-P) for t in [0,1], from P = (2, 2, 5) to Q = (-3, -5, -4). Write its y-coordinate as a function of t.

3.1 Recognizing Common Space Curves

  1. Worked exampleSketch and identify the curve traced by r(t) = (2cos t, 2sin t, t).
  2. Worked exampleFind where the curve r(t) = (t, t2, t3) crosses the plane z = 8.
  3. Going furtherWhat happens to the helix r(t) = (cos t, sin t, t) once the coefficient on t is doubled?
  4. Common mistakeWhy doesn't a curve that looks flat on a 2D screen have to actually lie in a single plane?
Your turn1 of 2
The curve r(t) = (t, t2, t3) crosses the plane z = 45 (with 45 > 0) at some t > 0. Find the y-coordinate at that point.

3.1 Limits and Continuity of Vector-Valued Functions

  1. How it worksHow do you take the limit of a vector-valued function by taking the limit of each component separately?
  2. Worked exampleFind the limit of r(t) = (sin t / t, t2 + 1, et) as t approaches 0.
  3. Common mistakeWhy doesn't every single component need to blow up for a vector-valued function to be discontinuous?
  4. Going furtherWhat happens to a vector-valued function's continuity at a value of t where just one component has a removable discontinuity?
Your turn1 of 2
Find the limit of the x-component of r(t) = (sin(4t)/t, t2+1, et) as t approaches 0.

3.2 Derivatives of Vector-Valued Functions

  1. How it worksHow does r'(t) come out tangent to the very curve that r(t) traces?
  2. Worked exampleFind r'(t) for r(t) = (t2, sin t, 3t).
  3. Worked exampleFind the tangent line to r(t) = (t, t2, t3) at t = 1.
  4. Common mistakeWhy isn't r'(t) itself a point that sits on the curve?
Your turn1 of 2
For r(t) = (t2, sin(t), 2t), write the y-component of r'(t) as a function of t.

3.2 Product and Chain Rules for Vector-Valued Functions

  1. How it worksHow do the dot product and cross product derivative rules mirror the ordinary product rule?
  2. Worked exampleFind d/dt of u(t) dot v(t) for u(t) = (t, 1, 0) and v(t) = (0, t, 2).
  3. Going furtherWhat changes about the derivative rule once a vector-valued function is composed with a scalar function of t?
  4. Worked exampleFind d/dt of u(t) x v(t) for u(t) = (t, 0, 1) and v(t) = (1, t, 0).
Your turn1 of 2
Find d/dt of u(t) dot v(t) for u(t) = (t, 3, 0) and v(t) = (0, t, 2).

3.2 The Unit Tangent Vector and Smooth Curves

  1. Worked exampleFind the unit tangent vector T(t) for r(t) = (3cos t, 3sin t, 4t).
  2. Common mistakeWhy isn't a curve smooth just because you can draw it without ever lifting your pen?
  3. Going furtherWhat happens to the unit tangent vector at a point where r'(t) equals the zero vector?
Your turn1 of 2
Find the z-component of the unit tangent vector T(t) for r(t) = (4cos t, 4sin t, 4t).

3.2 Integrals of Vector-Valued Functions

  1. How it worksHow does integrating a vector-valued function work term by term, the same way differentiating it does?
  2. Worked exampleFind the antiderivative of r(t) = (2t, cos t, et) satisfying r(0) = (1, 0, 1).
  3. Going furtherWhat changes about reconstructing a path from a velocity function when the initial position is unknown?
  4. Common mistakeWhy doesn't integrating a vector-valued function's magnitude give you back the length of the path it traces?
Your turn1 of 2
Find the antiderivative R(t) of r(t) = (4t, cos t, et) satisfying R(0) = (2, 5, -3). Write the x-component of R(t) as a function of t.

3.3 Arc Length of a Space Curve

  1. How it worksHow does integrating the magnitude of r'(t) add up the length of a curving path?
  2. Worked exampleFind the arc length of r(t) = (3cos t, 3sin t, 4t) from t = 0 to t = 2pi.
  3. Common mistakeWhy isn't arc length the same thing as the straight-line distance between a curve's two endpoints?
  4. Going furtherWhat happens to the arc length formula when a curve is only piecewise smooth?
Your turn1 of 2
Find the arc length of r(t) = (4cos t, 4sin t, 4t) from 0 to t, as a function of the upper limit t.

3.3 Arc Length Parametrization

  1. How it worksHow does inverting the arc length function s(t) reparametrize a curve by arc length?
  2. Worked exampleFind the arc length parametrization of the helix r(t) = (cos t, sin t, t) starting at t = 0.
  3. Going furtherWhat happens to the tangent vector's magnitude once a curve is parametrized by arc length?
Your turn1 of 2
For the helix r(t) = (cos t, sin t, 4t), the arc length from t=0 is s(t) = t*√(1+42). Write t as a function of arc length s.

3.3 The Osculating Circle and Radius of Curvature

  1. How it worksHow does the radius of curvature come out as simply 1 divided by the curvature at that point?
  2. Worked exampleFind the radius of curvature of y = x2 at the point (1, 1).
  3. Ties togetherHow does a tight osculating circle at a point connect to that point feeling like a sharper turn?
Your turn1 of 2
Find the radius of curvature of y = 4x2, as a function of x.

3.3 Curvature

  1. How it worksHow does the rate at which the unit tangent vector turns measure how sharply a curve bends?
  2. Worked exampleFind the curvature of r(t) = (t, t2, 0) at t = 1.
  3. Worked exampleFind the curvature of a circle of radius 5 directly from its formula.
  4. Common mistakeWhy doesn't a curve's curvature depend on how fast you happen to move along it?
Your turn1 of 2
Find the curvature of r(t) = (t, 4t2, 0), as a function of t.

3.3 The Cross Product Formula for Curvature

  1. How it worksHow does the formula kappa = |r' x r''| / |r'|3 avoid ever reparametrizing by arc length?
  2. Worked exampleFind the curvature of r(t) = (t, t2, t3) at t = 0 using the cross product formula.
  3. Going furtherWhat happens to this curvature formula the moment r'(t) and r''(t) point in exactly the same direction?
  4. Common mistakeWhy doesn't a straight line's curvature formula ever give anything other than zero?
Your turn1 of 2
Find the curvature of r(t) = (t, 4t2, 2t3) at t = 0, using kappa = |r' x r''| / |r'|3.

3.3 The Principal Unit Normal Vector and Osculating Plane

  1. How it worksHow is the principal unit normal vector built by differentiating and normalizing the unit tangent?
  2. Worked exampleFind the principal unit normal vector N(t) for r(t) = (cos t, sin t, 0).
  3. Ties togetherHow do the tangent and normal vectors at a point define the osculating plane that best fits the curve there?
Your turn1 of 2
For r(t) = (4cos t, 4sin t, 0), write the x-component of the principal unit normal vector N(t) as a function of t.

3.3 The Binormal Vector and the TNB Frame

  1. How it worksHow does crossing the tangent and normal vectors produce a binormal vector perpendicular to both?
  2. Worked exampleFind the binormal vector B(t) for r(t) = (cos t, sin t, t) at t = 0.
  3. Common mistakeWhy isn't the TNB frame a fixed set of axes the way i, j, and k is?
  4. Ties togetherHow does the TNB frame ride along a roller coaster car the same way it sits on any moving curve?
Your turn1 of 2
For the helix r(t) = (cos t, sin t, 3.63t), find the z-component of the binormal vector B(t).

3.4 Velocity and Acceleration in Space

  1. How it worksHow do position, velocity, and acceleration relate as a vector-valued function and its two derivatives?
  2. Worked exampleFind the velocity and acceleration of a particle with position r(t) = (t2, t3, 2t) at t = 1.
  3. Worked exampleFind the speed of a particle with position r(t) = (3cos t, 3sin t, 4t) at t = π/2.
  4. Common mistakeWhy doesn't moving at constant speed mean an object's acceleration is zero?
Your turn1 of 2
Find the y-component of the acceleration vector for r(t) = (t2, 4t3, 2t), as a function of t.

3.4 Projectile Motion in Three Dimensions

  1. How it worksHow does integrating a constant gravitational acceleration twice rebuild a projectile's full position vector?
  2. Worked exampleFind the position vector of a projectile launched from the origin with initial velocity (10, 5, 20) under gravity -9.8 m/s2 in the z-direction.
  3. Worked exampleFind how long the projectile from the previous setup takes to return to height z = 0.
Your turn1 of 2
A projectile launches from the origin with initial velocity (4, 3, 29) m/s under gravity -9.8 m/s2 in the z-direction. Write its z-coordinate as a function of time t.

3.4 Tangential and Normal Components of Acceleration

  1. How it worksHow do aT and aN project the acceleration vector onto the tangent and normal directions?
  2. Worked exampleFind the tangential and normal components of acceleration for r(t) = (t, t2, 0) at t = 1.
  3. Going furtherWhat happens to aN for an object moving at constant speed around a circular track?
  4. Common mistakeWhy doesn't aT alone tell you anything about how sharply an object is turning?
Your turn1 of 2
Find the tangential component of acceleration aT for r(t) = (t, 4t2, 0), as a function of t.

3.4 Kepler's Laws and Orbital Motion

  1. How it worksHow does conserving the cross product r(t) x v(t) force a planet's orbit to obey Kepler's second law?
  2. Worked exampleFind the areal velocity of a planet whose position and velocity satisfy r(t) x v(t) = (0, 0, 6).
  3. Ties togetherHow does Kepler's second law connect back to the cross product's role in measuring swept-out area?
Your turn1 of 2
A planet's position and velocity satisfy r(t) x v(t) = (0, 0, 11). Find its areal velocity, (1/2)|r(t) x v(t)|.

3.3 Torsion and the Frenet-Serret Formulas

  1. How it worksHow does torsion measure the rate the binormal vector tips away from a fixed osculating plane?
  2. Worked exampleFind the torsion of the helix r(t) = (cos t, sin t, t).
  3. Common mistakeWhy isn't a curve automatically planar just because its torsion happens to be zero at one single point?
  4. Ties togetherHow do the Frenet-Serret formulas package curvature and torsion into the rates T, N, and B change as you move along a curve?
Your turn1 of 2
Find the torsion of the helix r(t) = (cos t, sin t, 2.94t).