Name: vector functions in motion and displacement
Uploaded: 2026-06-08T09:38:47.452Z
Duration: 1 min 23 s
Channel: JoVE

Consider a drone moving through space with a velocity that varies over time. Velocity can be expressed as a vector-valued function, in which each scalar component is a real-valued function describing motion in a specific direction.In two-dimensional space, the drone&#x2019;s motion can be represented by horizontal and vertical components of the velocity function. The indefinite integral of the function is obtained by integrating each component independently, resulting in a new vector function that represents position at any time t.In definite integration, limits are applied to each scalar component, yielding the net change in position along the x-axis and the y-axis over the specified time interval.In three-dimensional space, an additional component accounts for motion along the third axis.Each component of the velocity function is integrated independently. Through this process, the complete position function in three dimensions is determined.When the vector-valued velocity function of the drone is integrated over definite limits, the result is a new vector representing the total displacement of the drone in all three spatial directions.

integrals of vector functions

Vector-valued functions provide a convenient framework for describing motion in space when both magnitude and direction are important. A drone&#x2019;s velocity at any instant has a direction and a speed, and as the drone moves, both can change. A vector-valued function captures this behavior by assigning to each time a vector whose components are real-valued functions. Each component represents motion along a particular axis in space. Such functions can describe motion in either two-dimensional or three-dimensional space, depending on the number of components included in the vector.In two dimensions, the velocity of the drone is described by two component functions: one for horizontal motion and one for vertical motion. To find the drone&#x2019;s position at any time, the velocity function is integrated with respect to time. This is done by integrating each component function separately. The result is a new vector function whose components give the horizontal and vertical positions of the drone at time t. The constant of integration appears as a constant vector, which corresponds to the drone&#x2019;s initial position at the reference time. The indefinite integral represents the family of possible position functions associated with a given velocity function. Physically, it describes the position of the object at any instant in time, together with information about its initial position.When the net change in position over a time interval is required, definite integration is used. By applying the limits of integration to each component function, the total change in horizontal and vertical position during the interval is obtained. These values form a vector that represents the drone&#x2019;s net displacement in the plane. The displacement vector indicates both the direction and the distance of the overall change in position. The definite integral represents the total displacement of the object over a specified time interval. Physically, it measures the overall change in position between the initial and final times rather than the position at an individual instant.In three-dimensional space, motion includes an additional component corresponding to the third axis. The velocity function, therefore, has three components, representing motion along each spatial direction. As in two dimensions, each component is integrated independently. The resulting position function combines all three position components and describes the drone&#x2019;s motion throughout space. When evaluated over a specific time interval, the integral produces a displacement vector representing the drone&#x2019;s total change in position in three dimensions.

Integrals of Vector Functions

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Education

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JoVE Core

Calculus

미적분학

Unit

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Vector Functions and Motion

벡터 함수와 운동

KEY TERMS AND CONCEPTS

핵심 용어 및 개념

벡터 함수의 적분

vector functions in motion and displacement

Vector functions describe motion in space when both speed and direction matter. A drone is a clear example, because its velocity changes as it moves. A vector-valued function assigns a vector to each time value. Each component is a real-valued function that tracks motion along one axis in space.

These functions can model motion in two dimensions or three dimensions. In two dimensions, the velocity of the drone has two component functions. One component shows horizontal motion, and the other shows vertical motion. In three dimensions, there is an additional component for the third axis, so the motion can be described in full space.

To find position from velocity, the velocity function is integrated with respect to time. Each component is integrated separately. The result is a new vector function whose components give the position along each axis at time t. The constant of integration appears as a constant vector. This constant vector represents the drone’s initial position at the reference time.

The indefinite integral gives the family of possible position functions for a given velocity function. In physical terms, it shows the object’s position at any instant together with its starting point. When the net change in position over a time interval is needed, definite integration is used instead. The limits of integration are applied to each component, and the result is a displacement vector in the plane or in space. This vector shows both the direction and the distance of the overall change in position.

Vector Functions in Motion and Displacement

vector-functions-in-motion-and-displacement

introduction to vector functions

Vector Functions for Motion in Space

Vector functions describe motion in space by assigning a vector to each input value. In three dimensions, a vector function is often written as r(t) = . The component functions f(t), g(t), and h(t) give the position along the x-, y-, and z-axes. The variable t usually stands for time. That makes r(t) a position vector for a moving object. As t changes, the position vector traces a space curve C, which is the path of a point in three-dimensional space. The terminal points P = (f(t), g(t), h(t))...

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Helices and Motion in Space Curves

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Using Space Curves to Predict Collisions

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derivatives of vector functions

Vector Motion and Instantaneous Velocity

Vector-valued functions describe position as a function of time. In Cartesian coordinates, a moving car on a curved road can be written as \begin{equation*} \textbf{r}(t)=\langle x(t),y(t),z(t)\rangle \end{equation*}. Each component gives the car’s location along one axis. The secant vector shows the change in position over a time interval. At time \(t\), the car is at point \(P\) with position \(\textbf{r}(t)\). After a short time \(h\), it reaches point \(Q\) with position...

Vector functions describe motion in space when both speed and direction matter. A drone is a clear example, because its velocity changes as it moves. A vector-valued function assigns a vector to each time value. Each component is a real-valued function that tracks motion along one axis in space. These functions can model motion in two dimensions or three dimensions. In two dimensions, the velocity of the drone has two component functions. One component shows horizontal motion, and the other...

arc length of space curves

Distance Traveled Along a Space Path

Arc length gives the total distance traveled along a curve in space. For a moving object such as a helicopter, the path can be written as a vector-valued position function r(t) = , where t is time. This distance is not the same as displacement, which measures only the straight-line change between two points. Arc length includes every change in direction along the route. To find this length, the motion interval is split into many small time steps of size Δt. Over each step, the curve is treated...

curvature and its interpretation

How Curves Change Direction

Curvature measures how fast a curve changes direction at a point. A small curvature means the curve bends gently. A large curvature means it turns more sharply. For a space curve, the position of a moving object can be written as a vector-valued function r(t), where t often stands for time. The direction of motion comes from the tangent vector. The unit tangent vector is the tangent vector scaled to length 1, and it shows the instantaneous direction of travel along the curve. Curvature tells...

the normal and binormal vectors

Frenet Frame on a Helical Track

A helical track shows how tangent, normal, and binormal vectors describe motion in space. As a roller coaster car moves along the spiral, these three unit vectors stay perpendicular to one another. Together, they form the Frenet-Serret frame, a moving coordinate system for a curve in three dimensions. The unit tangent vector shows the direction the car is moving at that exact point. It comes from the first derivative of the position vector with respect to arc length. The unit normal vector...

torsion in vector calculus

Twisting Space Curves with Torsion

Torsion describes how a space curve twists out of the plane of bending. It works alongside curvature, which measures how sharply the path bends. Together, these ideas help describe motion in three dimensions. The Frenet–Serret frame gives a local picture of a curve at each point. It uses three unit vectors. The tangent vector points in the direction of motion, the normal vector points toward the center of curvature, and the binormal vector comes from the cross product of the tangent and normal...

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motion-in-space-velocity-and-acceleration

Position Vectors, Speed, and Motion Changes

Motion in space can be described with a position vector r(t). This vector gives the location of an object, such as a drone moving through the air, at any time t. To study the motion, we look at how the position vector changes over time. Average velocity is found by dividing the change in position by the length of the time interval. When the interval becomes very small, this average velocity approaches a limit. That limit is the derivative of the position vector with respect to time. The...

projectile motion

Parabolic Paths in Projectile Motion

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tangential and normal components of acceleration

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Curved Motion and Acceleration Vectors

Curved motion changes acceleration in two parts: one along the path and one across it. In particle motion, acceleration is often split into tangential and normal components so the change in velocity is easier to understand. The tangential direction follows the path of motion. It shows how the particle’s speed changes over time. The normal direction points toward the center of curvature, or the center of the bend in the path. It shows how the direction of motion changes. A particle moving along...

vector functions and motion: problem solving

vector-functions-and-motion-problem-solving

Drone Motion from Acceleration to Position

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key terms and concepts

Learn how vector-valued functions model drone motion, how integrating each component gives position, and how definite integrals find displacement.

Vector Functions in Motion and Displacement | JoVE K12

The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Vector Functions in Motion and Displacement

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