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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JoVE Core

Calculus

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Eenheid

Vectors in Space

Vectoren in de ruimte

KEY TERMS AND CONCEPTS

BELANGRIJKE TERMEN EN CONCEPTEN

Inleiding tot Vectoren

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,...

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...

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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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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...

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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

Finding Cable Tensions in 3D Equilibrium

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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