Name: adding and multiplying vectors
Uploaded: 2023-09-22T11:49:19.000Z
Duration: 1 min 20 s
Channel: JoVE

Рассмотрим два вектора силы в двумерной системе координат.
Эти векторы могут быть сложены с помощью закона параллелограмма, при котором хвосты параллельных векторов соединяются вместе, а линии, параллельные каждому вектору, рисуются для формирования параллелограмма.
Результирующий вектор можно получить, проведя диагональ от хвостов до точки пересечения линий.
Аналогичным образом, векторы могут быть сложены с помощью правила треугольника, где два вектора расположены как две стороны треугольника, соединенные голова к хвосту, а третья сторона треугольника представляет величину и направление результирующего вектора.
Вычитание вектора похоже на сложение обратного одного вектора к другому. В результате разница между двумя векторами получается по одним и тем же правилам сложения векторов.
Когда вектор умножается на скаляр, величина вектора изменяется на величину скалярной величины.
Если скаляр положительный, то направление вектора остается прежним, а если отрицательное, то направление меняется на противоположное.

<ol>
	<li>Хиббелер, Р.К. (2016). Инженерная механика \u2012 Статика и динамика. Хобокен, Нью-Джерси: Пирсон Прентис Холл. С. 18 -19</li>
</ol>

Операции с векторами

Векторы — это физические величины, имеющие и величину, и направление. Операции с векторами включают сложение, вычитание и скалярное умножение.

Вектор, умноженный на скалярное значение, называется скалярным умножением. Полученный результат — это новый вектор с другой величиной. Если скаляр положительный, направление вектора остаётся таким же, но если он отрицательный, направление вектора меняется. Например, произведение массы на скорость даёт импульс. Здесь скалярная величина массы умножается на векторную величину скорости, что приводит к векторному количеству импульса. Направление полученного вектора импульса совпадает с направлением вектора скорости.

Два вектора могут быть сложены геометрически с использованием правила параллелограмма сложения векторов. При сложении с использованием этого правила два вектора представляются в качестве смежных сторон параллелограмма. Начала обоих векторов помещаются в начало координат, и проводятся параллельные линии к каждому вектору, чтобы образовать параллелограмм. Затем проводится диагональ параллелограмма от начала координат к противоположному углу параллелограмма, чтобы представить результирующий вектор. Величина и направление полученного вектора можно определить, измерив длину и углы диагонали параллелограмма. Правило параллелограмма может быть упрощено в особом случае, когда два вектора коллинеарные, то есть имеют одну линию действия. В этом случае результирующий вектор можно получить алгебраическим или скалярным сложением двух векторов.

Другой метод графического сложения двух векторов — это правило треугольника сложения векторов. Правило треугольника сложения векторов утверждает, что если два вектора расположены головой к хвосту в качестве двух сторон треугольника, третья сторона треугольника представляет величину и направление результирующего вектора. Сложение векторов с использованием этого правила включает размещение первого вектора в начале координат и размещение второго вектора головка к хвосту с первым вектором. Третья сторона треугольника от начала координат до конца второго вектора представляет результирующий вектор.

Процесс вычитания вектора включает сложение обратного (отрицательного) одного вектора к другому вектору. Это означает, что правила сложения векторов могут быть применены для получения разности между двумя векторами.

Векторы — это физические величины, которые имеют как величину, так и направление. Векторные операции включают сложение, вычитание и скалярное умножение.
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Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

Force Vectors

Векторы силы

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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 velocity is the vector quantity. The result is momentum, and its direction matches the direction of the velocity vector.

Two vectors can be added geometrically with the parallelogram law. In this method, the two vectors are drawn as adjacent sides of a parallelogram with both tails at the origin. Lines are then drawn parallel to each vector to complete the shape.

The diagonal of the parallelogram, drawn from the origin to the opposite corner, shows the resultant vector. Its magnitude and direction can be found by measuring the diagonal's length and angles. When the two vectors are collinear, meaning they share the same line of action, the resultant can be found by algebraic or scalar addition.

Another way to add vectors is the triangle law of vector addition. Here, the vectors are arranged head to tail as two sides of a triangle. The third side of the triangle, from the origin to the end of the second vector, represents the resultant vector.

Vector subtraction uses the reverse, or negative, of one vector added to another vector. This lets the rules of vector addition be used to find the difference between two vectors.

Adding and Multiplying Vectors

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

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...

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force classification

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vector addition of forces

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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

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scalar notation

Vector Components for Force Calculations

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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

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Resolving Forces in 3D Coordinates

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position vectors

Using Position Vectors to Locate Points

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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: problem solving

Using the Dot Product to Solve Vector Questions

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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 vector addition, subtraction, and scalar multiplication work using parallelogram and triangle laws, plus a momentum example for high school science.

Adding and Multiplying Vectors | 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

Adding and Multiplying Vectors

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