Name: cross product direction and magnitude
Uploaded: 2023-09-22T11:49:20.000Z
Duration: 1 min 25 s
Channel: JoVE

Перекрестное произведение — это произведение двух различных векторов, результирующим которых является третий вектор.
Величина перекрестного произведения получается путем умножения величины обоих векторов и синуса угла между ними.
Здесь величина перекрестного произведения может быть умножена на единичный вектор, определяющий направление результирующего вектора.
Правило правой руки задает направление результирующего вектора. Если согнутые пальцы представляют направление от вектора А к вектору В, то направление большого пальца представляет направление результирующего вектора.
Результирующий вектор всегда перпендикулярен плоскости, содержащей векторы A и B.
Перекрестное произведение векторов некоммутативно. Результирующая перекрестного произведения вектора А с вектором В равна по величине, но противоположна по направлению перекрестному произведению вектора В с вектором А.
Однако перекрестное произведение подчиняется ассоциативным и распределительным законам.

<ol>
	<li>Хиббелер, Р.К. (2016). Инженерная механика \u2012 Статика и динамика. Хобокен, Нью-Джерси: Пирсон Прентис Холл. С. 125 \u2012 126.</li>
</ol>

Векторное произведение

Cross Product

Векторное произведение является фундаментальным понятием в векторной алгебре, которое является векторной операцией над двумя различными векторами для получения третьего вектора. В отличие от скалярного произведения, векторное произведение дает векторную величину, перпендикулярную обоим исходным векторам.

Величина векторного произведения определяется путем умножения величин обоих векторов и синуса угла между ними. Это означает, что больший угол между векторами приведет к большей величине векторного произведения.

<img alt="Equation 1" src="/files/ftp_upload/14251/14251_Equation_1v3.svg" style="text-align: center; height: 11px;" />

Направление результирующего вектора определяется правилом правой руки. Если мы изогнем пальцы правой руки от вектора A к вектору B, то направление большого пальца будет представлять направление результирующего вектора. Другими словами, направление векторного произведения перпендикулярно плоскости, содержащей векторы A и B.

Важно отметить, что векторное произведение векторов не коммутативно. Иными словами, результирующий вектор для векторного произведения вектора A с вектором B имеет равную по величине, но противоположную по направлению величину векторного произведения вектора B с вектором A. Однако векторное произведение подчиняется ассоциативному закону и дистрибутивным законам сложения. Это означает, что векторное произведение суммы векторов равно сумме векторных произведений каждого вектора.

В заключение, векторное произведение является основным понятием в векторной алгебре, которое имеет широкое применение в физике и инженерии, такое как вычисление момента силы относительно точки, вычисление момента силы относительно оси и вычисление углового момента тела относительно оси.

Перекрестное произведение — фундаментальное понятие в векторной алгебре, представляющее собой векторную операцию над двумя различными векторами для получения третьего вектора. В отличие от ...

Education

JoVE Core

Mechanical Engineering

Машиностроение

Unit 1

Раздел 1

Force Vectors

Векторы силы

KEY TERMS AND CONCEPTS

ОСНОВНЫЕ ТЕРМИНЫ И ПОНЯТИЯ

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 fingers of the right hand curl from vector A to vector B, the thumb shows the direction of the result.

The cross product points perpendicular to the plane containing vectors A and B. It is also non-commutative. That means A cross B has the same magnitude as B cross A, but the direction is opposite.

The cross product still follows the associative law and the distributive law of addition. So, the cross product of a sum of vectors equals the sum of the cross products of each vector. This idea is useful in physics and engineering. It is used to calculate the moment of a force about a point, torque about an axis, and angular momentum about an axis.

Cross Product Direction and Magnitude

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cross product

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 cross product creates a perpendicular vector, how to use the right-hand rule, and why its size depends on angle, torque, and angular momentum.

Cross Product Direction and Magnitude | 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.

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Cross Product Direction and Magnitude

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