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

球在空气中运动可以通过矢量来描述——即具有大小和方向的量。在三维空间中，向量可被可视化为从一个点指向另一个点的箭头。箭头的长度表示向量的大小，其方向表示向量的方向。为了用数值描述一个矢量，需记录其在每个轴上的位移，这通过分量来表示。分量是用尖括号表示的数值。例如，从原点指向点 (3, 2, 1) 的向量具有分量〈3, 2, 1〉。这表示沿 x 轴方向移动 3 个单位，沿 y 轴方向移动 2 个单位，以及沿 z 轴方向向上移动 1 个单位。由于该向量起始于原点，因此被称为位置向量。当一个向量连接两个不同的点 A 和 B 时，其分量表示位置的变化。这些分量通过将 B 的坐标减去 A 的坐标得到。这些值表示沿每个轴的净位移。最后，矢量的大小即为其总长度，可通过扩展至三维的勾股定理计算得出。

载体简介

Introduction to Vectors

矢量为描述三维空间中的运动提供了一个简洁的数学框架。对于一个运动的球体，位移和速度等物理量自然地以矢量形式表示，因为它们同时包含大小和方向。从几何上看，矢量可被视作从一点延伸到另一点的箭头，箭头的长度对应矢量的大小，其在空间中的取向表示方向。这种表示方法使得矢量在分析物体在空中运动时尤为有用，因为此类运动通常沿多个轴同时发生。三维空间中的向量通常用其在 x、y 和 z 轴方向上的分量来表示。这些分量用尖括号括起来，例如 〈x, y, z〉，用于指明向量在各个坐标方向上的延伸程度。例如，从原点指向点 (3, 2, 1) 的向量写作 〈3, 2, 1〉。这表示该向量在 x 方向上位移 3 个单位，在 y 方向上位移 2 个单位，在 z 方向上位移 1 个单位。由于该向量起始于原点，因此被称为位置向量。当一个向量连接两个不同的点 A(x₁, y₁, z₁) 和 B(x₂, y₂, z₂) 时，其分量描述了从点 A 到点 B 的位置净变化。该向量可通过将点 B 的坐标减去点 A 的坐标得到：\begin{equation*}\vec{AB} = \langle x_2 - x_1,; y_2 - y_1,; z_2 - z_1 \rangle\end{equation*}这些分量差异显示了沿每个轴的位移，是追踪从一个位置到另一个位置运动的基础。矢量的大小是其总长度。在三维空间中，使用扩展形式的勾股定理进行计算：\begin{equation*}|\mathbf{v}| = \bm{\sqrt{x^2 + y^2 + z^2}}\end{equation*}该数值给出了位移的总体大小，与方向无关。

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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

introduction to vectors

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vectors in engineering applications

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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 and its geometry

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...

lines in space

Vector Form for a Line in Space

A line in three-dimensional space can be described with a vector equation. This method uses a known point on the line and a direction vector. It is a useful way to model straight paths in space. The transcript uses a surveying laser beam as an example. A laser beam from a surveying instrument can be treated as a line across a construction site. The beam starts at a known point, called P₀, whose position vector is r₀. The beam also has a direction vector v, which is parallel to its path. Any...

planes in space

Plane Equations from a Point and Normal

A plane in three-dimensional space can be described by one point on the plane and a normal vector. The normal vector is perpendicular to the surface, and it sets the plane’s orientation. In geometry, this gives a clear way to model a flat surface in space. The transcript uses a sloped glass panel on a building façade as a real-world example. Let r₀ be the position vector of a known point on the panel, such as the base mounting point. Let n be the normal vector to the panel. Any other point r...

cylinders in three-dimensional space

Modeling Cylindrical Surfaces in 3D

Cylindrical surfaces in three-dimensional space come from moving a two-dimensional curve in a straight line. As the curve slides, it leaves behind translated copies that build a surface. These copies form parallel rulings, which are straight lines in the same fixed direction. This idea helps describe some 3D shapes with simple equations in Cartesian coordinates. A cylindrical surface is often shown by an equation that leaves out one of the three variables. For example, y = x^2 does not include...

quadric surfaces

Shapes of Quadric Surfaces in 3D

Quadric surfaces are smooth three-dimensional shapes described by second-degree equations in x, y, and z. The way the squared and linear terms are arranged determines the surface. The main quadric surfaces are ellipsoids, cones, paraboloids, and hyperboloids. An ellipsoid is a closed surface that appears when all three variables are squared with the same sign. It is bounded and symmetric about its principal axes. Cross-sections taken along those axes produce circles or ellipses. A standard...

vectors in 2d: problem solving

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vectors in space: problem solving

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Three-dimensional static equilibrium can be used to find the tension in cables that support a chandelier. In this setup, the chandelier weighs 1000 N and sits at the origin of a 3D coordinate system. Three ceiling anchor points are located above it, and each cable runs from the chandelier to one anchor point. To analyze the forces, the direction of each cable must first be described with a position vector. A position vector is a vector that points from the chandelier to an anchor point. These...

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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