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

クロス積は、2 つの異なるベクトルの積であり、その結果が 3 番目のベクトルです。
クロス積の大きさは、両方のベクトルの大きさとそれらの間の角度の正弦を掛けることによって得られます。
ここで、クロス積の大きさは、結果のベクトルの方向を指定する単位ベクトルと乗算できます。
右手の法則は、結果のベクトルの方向を示します。カールした指がベクトル A からベクトル B への方向を表している場合、親指の方向は結果のベクトルの方向を表します。
結果のベクトルは、ベクトル A と B を含む平面に対して常に垂直になります。
ベクトルの外積は非可換です。ベクトル A とベクトル B の外積の結果は、大きさは等しくなりますが、ベクトル B とベクトル A の外積の方向とは逆になります。
ただし、クロス積は結合法則と分配法則に従います。

<ol>
	<li>ヒベラー、RC(2016)。エンジニアリングメカニクス \u2012 静力学と力学。 ニュージャージー州ホーボーケン:ピアソンプレンティスホール。125 \u2012 126ページ。</li>
</ol>

クロス積

Cross Product

クロス積はベクトル代数における基本的な概念であり、2つの異なるベクトルに対して行われるベクトル演算で、3番目のベクトルを得るものです。スカラー積とは異なり、クロス積は元のベクトルと直交するベクトルの量として結果が得られます。

クロス積の大きさは、ベクトルの大きさとそれらの間の角度の正弦を掛け合わせることで得られます。つまり、ベクトル間の角度が大きいほど、クロス積の大きさも大きくなります。

<img alt="Equation 1" src="/files/ftp_upload/14251/14251_Equation_1v3.svg" style="text-align: center; height: 11px;" />

結果ベクトルの方向は、右手の法則を用いて決定されます。右手の親指をベクトルAからベクトルBへ巻き、親指の向きが結果ベクトルの向きを表します。言い換えれば、クロス積の方向はベクトルAとベクトルBを含む平面に垂直です。

ベクトルのクロス積は非可換であることに注意することが重要です。つまり、ベクトルAとベクトルBのクロス積の結果ベクトルは大きさは同じですが、方向は逆です。しかし、クロス積は結合法則と分配法則に従います。つまり、ベクトルの和のクロス積はそれぞれのベクトルのクロス積の和と等しくなります。

まとめると、クロス積はベクトル代数における基本的な概念であり、物理学や工学に広く応用されます。例えば、力のモーメントの計算、軸周りのトルクの計算、物体の軸周りの角運動量の計算などに利用されます。

外積は、ベクトル代数の基本的な概念であり、2 つの異なるベクトルに対するベクトル演算で 3 番目のベクトルを取得します。とは異なり...

Education

JoVE Core

Mechanical Engineering

機械工学

Unit 1

ユニット1

Force Vectors

力のベクトル

KEY TERMS AND CONCEPTS

主要用語と概念

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 機械工学は、平衡、力、トラス解析、構造力学などの重要なトピックをカバーする簡潔でわかりやすいアニメーション ビデオを提供します。実際のアプリケーションを探索し、科学者主導の実験を通じてこれらのコンセプトが実際に動作しているのを目撃してください。

mechanical engineering

Cross Product Direction and Magnitude

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