Calculators / Calculus 3 / Vector-Valued Function Calculator

Vector-Valued Function Calculator

Published on: June 21, 2026

This Vector-Valued Function Calculator helps you work with a vector-valued function in three-dimensional space. From the same input function, you can find the derivative, second derivative, speed, unit tangent vector, and integral by applying the relevant formula to each component. If a value of t is given, the calculator can also evaluate the results at that point. It is a simple way to check answers, understand the method clearly, and practise vector calculus step by step.

Step-by-step method

  1. Identify the vector-valued function r(t).
  2. Differentiate the full vector to get r′(t), then solve each component.
  3. Differentiate again to get r′′(t), then solve each component.
  4. Find the speed by taking the magnitude of r′(t).
  5. Find the unit tangent vector T(t) = r′(t) / |r′(t)|.
  6. Integrate each component to get the vector integral.
  7. If a value of t is given, evaluate the results at that t.

Formulas:

Derivative and second derivative

r′( t ) = ⟨ x′( t ), y′( t ), z′( t ) ⟩
r′′( t ) = ⟨ x′′( t ), y′′( t ), z′′( t ) ⟩

Speed and unit tangent

2D speed: |r′( t )| = √(x′(t))2 + (y′(t))2
3D speed: |r′( t )| = √(x′(t))2 + (y′(t))2 + (z′(t))2
T( t ) =
r′( t )
|r′( t )|

Integral of a vector function

∫r( t )dt = ⟨ ∫x( t )dt, ∫y( t )dt, ∫z( t )dt ⟩ + C

Example 1: r(t)=<t^2,sin(t),e^t>

Step 1 - Identify the vector-valued function.

In this problem: Read the given vector function.

r( t ) = ⟨ t2, sin(t), e(t) ⟩

Step 2A - Differentiate the full vector function.

In this problem: Show the derivative of each component before simplifying.

r′( t ) = ⟨
d
dt
(t2),
d
dt
(sin(t)),
d
dt
(e(t)) ⟩

Step 2B - Differentiate component 1.

In this problem: Differentiate with respect to t.

d
dt
(t2) = 2t

Step 2C - Differentiate component 2.

In this problem: Differentiate with respect to t.

d
dt
(sin(t)) = cos(t)

Step 2D - Differentiate component 3.

In this problem: Differentiate with respect to t.

d
dt
(e(t)) = e(t)

Step 2E - Combine into the derivative vector.

In this problem: Put the differentiated components into ⟨ … ⟩.

r′( t ) = ⟨ 2t, cos(t), e(t) ⟩

Step 3A - Differentiate the velocity vector.

In this problem: Show the derivative of each velocity component before simplifying.

r′′( t ) = ⟨
d
dt
(2t),
d
dt
(cos(t)),
d
dt
(e(t)) ⟩

Step 3B - Differentiate velocity component 1.

In this problem: Differentiate again with respect to t.

d
dt
(2t) = 2

Step 3C - Differentiate velocity component 2.

In this problem: Differentiate again with respect to t.

d
dt
(cos(t)) = −sin(t)

Step 3D - Differentiate velocity component 3.

In this problem: Differentiate again with respect to t.

d
dt
(e(t)) = e(t)

Step 3E - Combine into the second derivative vector.

In this problem: Put the differentiated components into ⟨ … ⟩.

r′′( t ) = ⟨ 2, −sin(t), e(t) ⟩

Step 4A - Use the speed formula.

In this problem: Square each velocity component, then add and take the square root.

|r′( t )| = √2t2 + cos(t)2 + e(t)2

Step 4B - Simplify the speed.

In this problem: Simplify the square root.

|r′( t )| = √cos2(t) + 4t2 + e(2t)

Step 5 - Use the unit tangent formula.

In this problem: Divide the velocity vector by the speed.

T( t ) =
⟨ 2t, cos(t), e(t) ⟩
√cos2(t) + 4t2 + e(2t)
= ⟨
2t
√cos2(t) + 4t2 + e(2t)
,
cos(t)
√cos2(t) + 4t2 + e(2t)
,
e(t)
√cos2(t) + 4t2 + e(2t)
⟩

Step 6A - Integrate the full vector function.

In this problem: Show the integral of each component before simplifying.

∫ r( t )dt = ⟨ ∫ t2 dt, ∫ sin(t) dt, ∫ e(t) dt ⟩

Step 6B - Integrate component 1.

In this problem: Integrate with respect to t.

∫ t2 dt =
t3
3

Step 6C - Integrate component 2.

In this problem: Integrate with respect to t.

∫ sin(t) dt = −cos(t)

Step 6D - Integrate component 3.

In this problem: Integrate with respect to t.

∫ e(t) dt = e(t)

Step 6E - Combine into the vector integral.

In this problem: Put the antiderivatives into ⟨ … ⟩.

∫ r( t )dt = ⟨
t3
3
, −cos(t), e(t) ⟩ + C

Final answer: Vector function results computed

Example 2: r(t)=<t^2,sin(t),e^t>; t=1

Step 1A - Identify the vector-valued function.

In this problem: Read the given vector function.

r( t ) = ⟨ t2, sin(t), e(t) ⟩

Step 1B - Identify the value of t.

In this problem: This value will be used for evaluation.

t = 1

Step 2A - Differentiate the full vector function.

In this problem: Show the derivative of each component before simplifying.

r′( t ) = ⟨
d
dt
(t2),
d
dt
(sin(t)),
d
dt
(e(t)) ⟩

Step 2B - Differentiate component 1.

In this problem: Differentiate with respect to t.

d
dt
(t2) = 2t

Step 2C - Differentiate component 2.

In this problem: Differentiate with respect to t.

d
dt
(sin(t)) = cos(t)

Step 2D - Differentiate component 3.

In this problem: Differentiate with respect to t.

d
dt
(e(t)) = e(t)

Step 2E - Combine into the derivative vector.

In this problem: Put the differentiated components into ⟨ … ⟩.

r′( t ) = ⟨ 2t, cos(t), e(t) ⟩

Step 3A - Differentiate the velocity vector.

In this problem: Show the derivative of each velocity component before simplifying.

r′′( t ) = ⟨
d
dt
(2t),
d
dt
(cos(t)),
d
dt
(e(t)) ⟩

Step 3B - Differentiate velocity component 1.

In this problem: Differentiate again with respect to t.

d
dt
(2t) = 2

Step 3C - Differentiate velocity component 2.

In this problem: Differentiate again with respect to t.

d
dt
(cos(t)) = −sin(t)

Step 3D - Differentiate velocity component 3.

In this problem: Differentiate again with respect to t.

d
dt
(e(t)) = e(t)

Step 3E - Combine into the second derivative vector.

In this problem: Put the differentiated components into ⟨ … ⟩.

r′′( t ) = ⟨ 2, −sin(t), e(t) ⟩

Step 4A - Use the speed formula.

In this problem: Square each velocity component, then add and take the square root.

|r′( t )| = √2t2 + cos(t)2 + e(t)2

Step 4B - Simplify the speed.

In this problem: Simplify the square root.

|r′( t )| = √cos2(t) + 4t2 + e(2t)

Step 5 - Use the unit tangent formula.

In this problem: Divide the velocity vector by the speed.

T( t ) =
⟨ 2t, cos(t), e(t) ⟩
√cos2(t) + 4t2 + e(2t)
= ⟨
2t
√cos2(t) + 4t2 + e(2t)
,
cos(t)
√cos2(t) + 4t2 + e(2t)
,
e(t)
√cos2(t) + 4t2 + e(2t)
⟩

Step 6A - Integrate the full vector function.

In this problem: Show the integral of each component before simplifying.

∫ r( t )dt = ⟨ ∫ t2 dt, ∫ sin(t) dt, ∫ e(t) dt ⟩

Step 6B - Integrate component 1.

In this problem: Integrate with respect to t.

∫ t2 dt =
t3
3

Step 6C - Integrate component 2.

In this problem: Integrate with respect to t.

∫ sin(t) dt = −cos(t)

Step 6D - Integrate component 3.

In this problem: Integrate with respect to t.

∫ e(t) dt = e(t)

Step 6E - Combine into the vector integral.

In this problem: Put the antiderivatives into ⟨ … ⟩.

∫ r( t )dt = ⟨
t3
3
, −cos(t), e(t) ⟩ + C

Step 7A - Evaluate the position vector.

In this problem: Substitute the given value of t.

r( 1 ) = ⟨ 1, 0.84, 2.72 ⟩

Step 7B - Evaluate the velocity vector.

In this problem: Substitute the given value of t.

r′( 1 ) = ⟨ 2, 0.54, 2.72 ⟩

Step 7C - Evaluate the acceleration vector.

In this problem: Substitute the given value of t.

r′′( 1 ) = ⟨ 2, -0.84, 2.72 ⟩

Step 7D - Evaluate the speed.

In this problem: Substitute the given value of t.

|r′( 1 )| = √4 + cos2( 1 ) + e( 2 ) ≈ 3.42

Step 7E - Evaluate the unit tangent vector.

In this problem: Substitute the given value of t.

T( 1 ) = ⟨ 0.59, 0.16, 0.80 ⟩

Final answer: Vector function results computed