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

任意の 2 つのベクトルの内積は、ベクトルの大きさとベクトル間の角度の余弦の積です。
物体がロープで地面に沿って車両に引っ張られているとします。ロープが水平軸と角度θを作る場合、物体を一定の距離だけ変位させるために加えられた力の内積を使用して、行われる作業を計算できます。
デカルト形式で表される任意の 2 つのベクトルの内積は、対応する x、y、z 成分を乗算し、それらの積を代数的に加算することで決定できます。
角度シータが不明な場合、内積を使用して、逆余弦関数を使用して 2 つのベクトル間の角度を決定できます。
変位の方向に沿った力成分は、ベクトルのドット積とその方向を定義する単位ベクトルを取ることによって決定できます。
ベクトルの内積は、常に可換法則と加算と乗算の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。70ページ</li>
</ol>

ドット積

ドット積は数学と物理学において重要な概念です。

エンジニアリングでは、任意の2つのベクトルのドット積は、ベクトルの大きさの積とそれらの間の角度の余弦によって計算されます。2つのベクトル間にドットシンボルが使われます。

地面にロープを使って物体を引っ張る車両を考えてみましょう。もしロープが水平軸と角度を成している場合、加えられた力と物体の変位のドット積を使って行われた仕事を計算することができます。

直交座標形式で表される2つのベクトルのドット積は、それらの対応するx、y、z成分をかけ合わせて代数学的に足し合わせることによって求めることができます。これはベクトルAとBのドット積の公式によって示されます:

<img alt="Equation 1" src="/files/ftp_upload/14238/14238_Equation_1.svg" style="text-align: center; height: 15px;" />

2つのベクトル間の角度が未知の場合、ドット積は逆余弦関数を使って角度を求めるのに役立ちます。AとBが2つのベクトルであり、その間の角度が次のように求められるとしましょう:

<img alt="Equation 1" src="/files/ftp_upload/14238/14238_Equation_2.svg" style="text-align: center; height: 30px;" />

ドット積は加法と乗法の可換法則と分配法則を遵守します。ドット積は物理学やエンジニアリングの多くの分野で使用される基本的な操作です。仕事の計算、ベクトル間の角度の求め方、特定の方向における力の成分の決定に使用されます。その応用の多さから、ドット積はベクトル代数で重要な概念です。

ドット積は、数学と物理学の本質的な概念です。
エンジニアリングでは、任意の 2 つのベクトルの内積は、...

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

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

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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

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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 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

Using Dot Product to Find Work and Angle

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