Name: vectors in three-dimensional motion
Uploaded: 2026-06-08T09:38:13.579Z
Duration: 1 min 29 s
Channel: JoVE

The motion of a ball through the air can be described using vectors&#x2014;quantities that have both magnitude and direction.In three-dimensional space, a vector is visualized as an arrow pointing from one point to another. The arrow&#x27;s length represents the vector&#x27;s magnitude, and its orientation shows directionTo describe a vector numerically, its displacement along each axis is captured using components. Components are numerical values written in angle brackets.For example, a vector from the origin to the point (3, 2, 1) has components &#x3008;3, 2, 1&#x3009;. This shows a displacement of three units along the x-axis, two units along the y-axis, and one unit upward along the z-axis. Since this vector starts at the origin, it is called a position vector.When a vector connects two distinct points, A and B, its components represent the change in position. These components are found by subtracting the coordinates of A from those of B. These values show the net displacement along each axis.Finally, the magnitude of a vector is its total length. It is calculated using the Pythagorean Theorem extended to three dimensions.

introduction to vectors

Vectors provide a concise mathematical framework for describing motion in three-dimensional space. For a moving ball, quantities such as displacement and velocity are naturally represented as vectors because they include both magnitude and direction. Geometrically, a vector is visualized as an arrow extending from one point to another. The length of the arrow corresponds to the vector&#x2019;s magnitude, while its spatial orientation shows direction. This representation makes vectors especially useful for analyzing motion through the air, where movement occurs simultaneously along multiple axes.A vector in three-dimensional space is commonly expressed in terms of its components along the x-, y-, and z-axes. These components are written in angle brackets, such as &#x3008;x, y, z&#x3009;, and specify the extent of the vector in each coordinate direction. For example, the vector from the origin to the point (3, 2, 1) is written as &#x3008;3, 2, 1&#x3009;. This means the displacement is 3 units in the x-direction, 2 units in the y-direction, and 1 unit in the z-direction. Because this vector begins at the origin, it is called a position vector.When a vector connects two distinct points, A(x&#x2081;, y&#x2081;, z&#x2081;) and B(x&#x2082;, y&#x2082;, z&#x2082;), its components describe the net change in position from A to B. The vector is found by subtracting the coordinates of A from those of B:\begin{equation*}\vec{AB} = \langle x_2 - x_1,; y_2 - y_1,; z_2 - z_1 \rangle\end{equation*}These component differences show the displacement along each axis and are fundamental in tracking motion from one location to another.The magnitude of a vector is its total length. In three dimensions, it is calculated using the Pythagorean Theorem in extended form:\begin{equation*}|\mathbf{v}| = \bm{\sqrt{x^2 + y^2 + z^2}}\end{equation*}This quantity gives the overall size of the displacement, independent of direction.

Introduction to Vectors

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Education

教育

JoVE Core

Calculus

微积分

Unit

单元

Vectors in Space

空间向量

KEY TERMS AND CONCEPTS

关键术语与概念

向量简介

vectors in three-dimensional motion

Vectors describe motion in three-dimensional space with both size and direction. They are useful for quantities such as displacement and velocity, since those measurements need more than just a number. A vector is shown as an arrow from one point to another. The arrow’s length gives the magnitude, and its direction shows the way the motion points.

In three dimensions, a vector is written in terms of its x-, y-, and z-components. These components are often shown with angle brackets, such as ⟨x, y, z⟩. For example, the vector from the origin to the point (3, 2, 1) is written as ⟨3, 2, 1⟩. This means the displacement is 3 units in the x-direction, 2 units in the y-direction, and 1 unit in the z-direction. Because it starts at the origin, this is called a position vector.

A vector can also connect two different points, A(x1, y1, z1) and B(x2, y2, z2). In that case, its components show the net change in position from A to B. The vector is found by subtracting the coordinates of A from those of B: \begin{equation*}\vec{AB} = \langle x_2 - x_1,; y_2 - y_1,; z_2 - z_1 \rangle\end{equation*} These differences give the displacement along each axis and help track motion from one location to another.

The magnitude of a vector is its total length. In three dimensions, it is found with the extended Pythagorean Theorem: \begin{equation*}|\mathbf{v}| = \bm{\sqrt{x^2 + y^2 + z^2}}\end{equation*} This value gives the overall size of the displacement without including direction.

Vectors in Three-Dimensional Motion

Vectors describe motion in three-dimensional space with both size and direction. They are useful for quantities such as displacement and velocity, since those measurements need more than just a number. A vector is shown as an arrow from one point to another. The arrow’s length gives the magnitude, and its direction shows the way the motion points. In three dimensions, a vector is written in terms of its x-, y-, and z-components. These components are often shown with angle brackets, such as ⟨x,...

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Cable tension and force balance are shown with a steel beam held by two identical cables. The beam has a downward weight of 5000 N. The cables support it from opposite sides. Because the setup is symmetric, each cable makes a 60° angle with the horizontal beam and carries the same tension. The beam stays completely at rest, so it is in static equilibrium. Static equilibrium means the total horizontal force and the total vertical force are both zero. Each cable pulls in its own direction, so...

dot product of two vectors

Vector Projection and the Dot Product

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Cross Product, Area, and Torque

The cross product connects vector direction, area, and rotation in three-dimensional space. When two non-zero vectors are not parallel, they define a unique plane and form a parallelogram. Their cross product gives a third vector that is perpendicular to that plane. The direction of this new vector is found with the Right-Hand Rule. Point the fingers of your right hand along the first vector and curl them toward the second vector. Your thumb shows the direction of the result. This convention...

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Vector Form for a Line in Space

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cylinders in three-dimensional space

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quadric surfaces

Shapes of Quadric Surfaces in 3D

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vectors in 2d: problem solving

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vectors in space: problem solving

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key terms and concepts

Learn how vectors show motion with magnitude and direction, use x-, y-, and z-components, find a position vector, and calculate vector magnitude in 3D.

Vectors in Three-Dimensional Motion | 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

Vectors in Three-Dimensional Motion

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