Name: using dot product to find work and angle
Uploaded: 2025-05-22T09:06:02.000Z
Duration: 1 min 29 s
Channel: JoVE

Скалярное произведение любых двух векторов является произведением величины векторов и косинуса угла между ними.
Представьте себе объект, который транспортное средство тянет по земле за веревку. Если канат делает угол тета с горизонтальной осью, то проделанную работу можно рассчитать с помощью точечного произведения силы, приложенной для смещения объекта на определенное расстояние.
Скалярное произведение любых двух векторов, выраженных в декартовой форме, может быть определено путем умножения соответствующих им компонентов x, y и z и алгебраического сложения их произведений.
Предположим, что угол тета неизвестен, скалярное произведение может быть использовано для определения угла между двумя векторами с помощью обратной косинусной функции.
Силовую составляющую вдоль направления смещения можно определить, взяв скалярное произведение вектора с единичным вектором, определяющим его направление.
Скалярное произведение векторов всегда подчиняется коммутативному закону и распределительному закону сложения и умножения.

<ol>
	<li>Хиббелер, Р.К. (2016). Инженерная механика \u2012 Статика и динамика. Хобокен, Нью-Джерси: Пирсон Прентис Холл. С. 70</li>
</ol>

Скалярное произведение

Скалярное произведение - это важное понятие в математике и физике.

В инженерии скалярное произведение любых двух векторов является произведением их модулей и косинуса угла между ними. Оно обозначается точкой между двумя векторами.

Рассмотрим транспортное средство, тянущее предмет по земле с помощью веревки. Если веревка образует угол с горизонтальной осью, работа, совершаемая, может быть определена с помощью скалярного произведения приложенной силы и перемещения предмета.

Скалярное произведение двух векторов в декартовой форме можно определить, умножив соответствующие компоненты x, y, и z и алгебраически сложив их произведения. Это задается формулой для скалярного произведения векторов A и B: 

<img alt="Equation 1" src="/files/ftp_upload/14238/14238_Equation_1.svg" style="text-align: center; height: 15px;" />

Если угол между двумя векторами неизвестен, скалярное произведение может помочь определить угол с помощью обратной функции косинуса. Предположим, что A и B - два вектора, и угол между ними может быть определен:

<img alt="Equation 1" src="/files/ftp_upload/14238/14238_Equation_2.svg" style="text-align: center; height: 30px;" />

Скалярное произведение подчиняется коммутативному и дистрибутивному законам сложения и умножения. Скалярное произведение является основной операцией, используемой во многих областях физики и инженерии. Оно может вычислять проделанную работу, находить угол между векторами и определять составляющие силы вдоль определенного направления. Благодаря своим многочисленным применениям, скалярное произведение является важным понятием в векторной алгебре.

Скалярное произведение является важным понятием в математике и физике.
В инженерии скалярное произведение любых двух векторов является произведением величин ...

Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

Force Vectors

Векторы силы

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

Dot Product

using dot product to find work and angle

The dot product connects vectors, work, and direction in mathematics and physics. In engineering, it is found by multiplying the magnitudes of two vectors and the cosine of the angle between them. It is shown with a dot symbol between the vectors.

A real-world example is a vehicle pulling an object along the ground with a rope. If the rope is angled above the horizontal axis, the work done can be calculated with the dot product of the applied force and the object’s displacement. This makes the dot product useful for measuring motion in a specific direction.

Vectors written in Cartesian form can also be handled with the dot product. You multiply the matching x, y, and z components and then add the results algebraically. The same formula works for vectors A and B in component form.

The dot product can also help find the angle between two vectors when that angle is unknown. By using the inverse cosine function, the angle can be determined from the dot product value. The operation follows the commutative and distributive laws, so it behaves consistently with addition and multiplication.

Because of these properties, the dot product is used in many areas of physics and engineering. It can calculate work done, find the angle between vectors, and determine force components along a chosen direction. For students learning vector algebra, it is a key tool with many applications.

Using Dot Product to Find Work and Angle

scalar and vectors

Scalar vs Vector Quantities in Mechanics

Scalar and vector quantities are two main ways to describe physical quantities in mechanics. Common examples include force, speed, velocity, and work. Some of these are scalars, and others are vectors. A scalar is a quantity that has magnitude only. It does not need direction. Mass, area, and length are examples of scalar quantities. Scalars with the same units can be added or subtracted like ordinary numbers. A class that ends 10 min earlier than 50 min lasts 50 min minus 10 min, which is 40...

vector operations

Adding and Multiplying Vectors

Vectors are quantities with both magnitude and direction, and their operations include addition, subtraction, and scalar multiplication. Scalar multiplication means a vector is multiplied by a scalar value, which creates a new vector with a different magnitude. If the scalar is positive, the direction stays the same. If the scalar is negative, the direction reverses. A common example is momentum, which comes from multiplying mass by velocity. In this case, mass is the scalar quantity and...

introduction to force

Force, Motion, and Mechanical Effects

Force changes the motion, shape, or orientation of an object. A flowing stream of water can push on a turbine vane and make it move in the direction of the flow. In mechanics, force is treated as a vector quantity, which means it has both magnitude and direction. It is measured in newtons, or N, in the SI unit system. Forces can act through direct contact or without contact. A push or pull is a contact force. Magnetic and gravitational forces are examples of non-contact interactions. For an...

force classification

Types of Forces in Physics

Forces in physics can be grouped by how they act and where they come from. These groups help describe motion, behavior, and equilibrium in the physical world. They are also useful in physics and engineering. One common way to classify forces is by whether they need contact. Contact forces act only when two objects touch. Frictional force, tension, and normal force are all contact forces that come from surface contact. Forces can also act without physical touch. These are called non-contact...

vector addition of forces

Resultant Force from Tugboats

Resultant force can be found with vector addition when two or more forces act on the same object. This idea helps explain how multiple pulls combine to create one net force. It is a key tool for understanding motion in physics. A clear example is a ship pulled by two small tugboats. The forces, F1 and F2, act at the same time but in different directions. The net force on the ship is the vector sum of those two forces. The parallelogram law gives a geometric way to add vectors. In this method,...

two-dimensional force system

Resolving Forces in a Plane

Resolving forces in a plane is a key part of two-dimensional mechanical analysis. A force vector can be split into x- and y-components, which show how the force acts in each direction. These components are written with the help of unit vectors, one for the x direction and one for the y direction. To find the components in a Cartesian coordinate system, first identify the angle theta between the force and the positive x-axis. If the line of action passes through the origin, basic trigonometry...

two-dimensional force system: problem solving

two-dimensional-force-system-problem-solving

Two-Dimensional Forces: Free-Body Analysis

Two-dimensional force analysis starts with a free-body diagram. A free-body diagram shows the object and every external force acting on it. It includes each force’s size, direction, and point of application. After the diagram is drawn, each force is broken into x- and y-components. This uses vector analysis and trigonometry, especially sine and cosine. The forces can then be written in Cartesian vector form, which makes them easier to work with. Next, the components in each direction are...

scalar notation

Vector Components for Force Calculations

Scalar notation helps simplify vector calculations by breaking a vector into separate x and y parts. When vectors are added or subtracted, each component can be handled on its own using scalar notation. This makes force problems easier to solve.

cartesian vector notation

Vector Components in 3D Space

Cartesian vector notation shows how to write vectors in three-dimensional space using x, y, and z components. It gives a clear way to represent direction and size for quantities used in mechanical engineering. The notation uses the unit vectors i, j, and k for the x, y, and z axes. A vector pointing in the direction (3, −4, 5) can be written as A = 3i - 4j + 5k. The unit vectors are i = (1, 0, 0), j = (0, 1, 0), and k = (0, 0, 1). This form is useful for describing displacement, velocity,...

direction cosines of a vector

Vector Orientation with Direction Cosines

Direction cosines describe how a vector is oriented relative to the x, y, and z axes. For a vector A written in Cartesian form with i, j, and k unit vectors, this idea helps connect the vector’s components to its direction in space. The magnitude of vector A is the square root of the sum of the squares of its components. Its direction is described by the coordinate direction angles α, β, and γ with the x, y, and z axes. These angles can be found by projecting vector A onto each axis. The...

three-dimensional force system

Force Components in 3D Space

A three-dimensional force system describes forces acting in x, y, and z directions. In mechanical engineering, this framework helps students and engineers study how objects and structures behave in three-dimensional space. One common example is a force acting on a particle in space. That force can be split into three parts: the x-component, the y-component, and the z-component. Vector mathematics is used to calculate each component, and together they make up the full three-dimensional force...

three-dimensional force system:problem solving

three-dimensional-force-systemproblem-solving

Resolving Forces in 3D Coordinates

Three-dimensional force systems involve forces acting in three different directions at the same time. These problems are common in physics and engineering. They are used to find the resultant force, which helps predict or analyze how an object or structure will behave. The first step is to resolve each force into scalar components. A scalar component is the amount of force along one axis. Trigonometric functions and vector addition are used to break the force into parts and then combine those...

position vectors

Using Position Vectors to Locate Points

Position vectors are used to locate points in space and show the direction and distance between two points. In math and science, they help describe spatial relationships and make calculations easier. A position vector can connect one point to another point or show a point’s location relative to the origin of coordinates. For a point P(x, y, z) relative to the origin O, the position vector r runs from O to P. In Cartesian vector form, it is written as r = x i + y j + z k. Here, i, j, and k are...

force vector along a line

Finding a Cable Force from Two Points

A cable force can be found from the two points that define its line of action. In three-dimensional statics, this setup often appears with a pole and a cable anchored to the ground. The cable ends are labeled A and B, and their position vectors are written as P_A and P_B. The position vector from A to B is found using the triangle law of vector addition. First, subtract P_A from P_B to get P_AB. Then find the magnitude of P_AB by taking the square root of the sum of the squares of its...

dot product

The dot product connects vectors, work, and direction in mathematics and physics. In engineering, it is found by multiplying the magnitudes of two vectors and the cosine of the angle between them. It is shown with a dot symbol between the vectors. A real-world example is a vehicle pulling an object along the ground with a rope. If the rope is angled above the horizontal axis, the work done can be calculated with the dot product of the applied force and the object’s displacement. This makes the...

dot product: problem solving

Using the Dot Product to Solve Vector Questions

The dot product helps solve vector questions by combining vector size and the cosine of the angle between them. It is useful when the angle is measured anti-clockwise and when a problem asks for a vector relationship in component form. A good first step is to identify the question being asked. Then list the given vectors and write them in Cartesian form, also called component form. This makes the data easier to use in a calculation. Next, decide whether the dot product is the right operation.

cross product

Cross Product Direction and Magnitude

The cross product is a vector operation in vector algebra that creates a third vector from two input vectors. Unlike the scalar product, it does not give a single number. The result is a vector perpendicular to both original vectors. The size, or magnitude, of the cross product depends on the magnitudes of both vectors and the sine of the angle between them. A larger angle between the vectors gives a larger cross product magnitude. The direction is found with the right-hand rule. If the...

key terms and concepts

Learn how the dot product helps calculate work, find angles between vectors, and break force into components using vector algebra and Cartesian form.

Using Dot Product to Find Work and Angle | JoVE K12

JoVE Core Mechanical Engineering offers concise, easy-to-understand animated videos covering key topics such as equilibrium, forces, truss analysis, and structural mechanics. Explore their real-world applications, and witness these concepts in action through scientist-led experiments.

mechanical engineering

Using Dot Product to Find Work and Angle

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