Mathfolis

Vector-Valued Functions

Unit 4 · Functions Involving Parameters, Vectors, and Matrices

What AP Precalc asks here

A vector-valued function is the same idea as a parametric function, written in vector notation: r⃗(t)=(x(t),y(t))\vec r(t) = (x(t), y(t)) or equivalently r⃗(t)=x(t)i^+y(t)j^\vec r(t) = x(t) \hat i + y(t) \hat j. As tt varies, r⃗(t)\vec r(t) traces a curve in the plane. Evaluation is component-wise; curve identification uses parameter elimination just like in Topic 4.1.

Notation

Component form
r⃗(t)=(x(t),y(t))\vec r(t) = (x(t), y(t))
i, j form
r⃗(t)=x(t)i^+y(t)j^\vec r(t) = x(t) \hat i + y(t) \hat j
AP Tip: Vector-valued functions and parametric functions describe the same thing. The vector notation is just a tidy way to write 'both coordinates at once'.
Type 1

Evaluate r(t)

Substitute tt into both components.

Example 1
For r⃗(t)=(t2−1,3t)\vec r(t) = (t^2 - 1, 3t), find r⃗(2)\vec r(2). (A) (3,6)(3, 6) (B) (5,6)(5, 6) (C) (3,5)(3, 5) (D) (1,6)(1, 6)

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Type 2

Identify the traced curve

Eliminate tt from the components.

Example 2
Identify the curve traced by r⃗(t)=(3cos⁡t,3sin⁡t)\vec r(t) = (3 \cos t, 3 \sin t). (A) Line (B) Parabola (C) Circle of radius 3 centered at origin (D) Ellipse with semi-axes 3 and 1

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Type 3

Decompose into i, j components

Switch between (x(t),y(t))(x(t), y(t)) and x(t)i^+y(t)j^x(t) \hat i + y(t) \hat j forms.

Example 3
Write r⃗(t)=2ti^+(t−1)j^\vec r(t) = 2t \hat i + (t - 1) \hat j in component form. (A) (2t,t−1)(2t, t - 1) (B) (t−1,2t)(t - 1, 2t) (C) (2t+1,t)(2t + 1, t) (D) (t,t−1)(t, t - 1)

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