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9.4 Defining and Differentiating Vector-Valued Functions

Syllabus
2020
Topic
9.4
Level
—

Differentiate a Vector One Component at a Time

A planar vector-valued function r(t)=⟨x(t),y(t)⟩\mathbf r(t)=\langle x(t),y(t)\rangle gives both coordinates of a moving point at the same parameter value. Its derivative records the instantaneous horizontal and vertical rates together, so it points tangent to the path when it is nonzero.

\mathbf r'(t)=\left\langle x'(t),y'(t)\right\rangle

  1. Keep the component order fixed.\n2. Differentiate each component with respect to the same parameter.\n3. Reassemble the derivatives inside vector brackets.\n4. Substitute a parameter value componentwise when an instantaneous vector is required.

Example: r(t)=⟨t2−1,t3+2t⟩\mathbf r(t)=\langle t^2-1,t^3+2t\rangle. Then r′(t)=⟨2t,3t2+2⟩\mathbf r'(t)=\langle2t,3t^2+2\rangle. At t=1t=1, the point is r(1)=⟨0,3⟩\mathbf r(1)=\langle0,3\rangle and the derivative is r′(1)=⟨2,5⟩\mathbf r'(1)=\langle2,5\rangle, a tangent direction and, for a position function, the instantaneous velocity vector.

The derivative is a vector, not the sum or magnitude of the component derivatives. Its magnitude ∥r′(t)∥=[x′(t)]2+[y′(t)]2\|\mathbf r'(t)\|=\sqrt{[x'(t)]^2+[y'(t)]^2} is a separate scalar quantity; for motion, that magnitude is speed.