Name: using position vectors to locate points
Uploaded: 2025-05-22T09:06:01.000Z
Duration: 1 min 29 s
Channel: JoVE

位置ベクトルは、空間内の点の位置を別の点に対して相対的に位置付ける固定ベクトルです。
壁に固定されたネジ付きアイとそれに取り付けられたケーブルOPを考えてみましょう。原点にねじの目、軸に沿ったコンポーネントベクトルを使用して、ケーブル上の点Pを配置する直交座標系を確立します。
原点から点Pに向けられた位置ベクトルrは、デカルトベクトル形式で表すことができます。
デカルト ベクトルの一般的な表現として、壁に固定されたブラケットとそれに取り付けられたコードについて考えてみます。
nA と nB を原点から点 A と B まで伸びる位置ベクトルとします。
したがって、点AからBへの結果の位置ベクトルnは、三角形の法則を使用し、nAとnbをデカルトベクトル形式で表すことによって取得できます。
i、j、および k は、B から A の座標を減算して形成される位置ベクトル n の成分に沿った単位ベクトルです。

<ol>
	<li>ラッセルC.ヒベラー。(2016). エンジニアリングメカニクス:静力学とダイナミクス。ピアソン。(ページ番号 – 56,57)</li>
</ol>

位置ベクトル

位置ベクトルは数学の基本的な概念であり、空間内のある点の位置を他の点に関して決定するのに役立ちます。それは2つの点の方向と距離を記述するベクトルです。位置ベクトルは数学や科学の分野で非常に有用であり、空間的な関係を表現し、計算を容易にするのに役立ちます。

例えば、座標の原点に対して点P(x, y, z)の位置を特定したい場合、原点Oから点Pまで延びる位置ベクトルrを定義することができます。このベクトルは、Cartesian座標系のベクトル形式で以下のように表すことができます：r = x*i + y*j + z*k。ここで、i、j、kはx、y、z方向の単位ベクトルです。位置ベクトルrは、点Oから点Pまでのベクトルの方向と大きさを与えてくれます。

空間内の点Aから点Bへ向かう位置ベクトルを考えましょう。このベクトルは記号rで示すことができます。また、ベクトルが向かう元の点と先の点を示すために、2つの添え字を使ってこのベクトルを指定することもできます。したがって、rをrABとして示すこともできます。ただし、位置ベクトルが座標の原点から延びる場合は、rAとrBという1つの添え字だけを使って指定します。位置ベクトルrABは、式r_AB = rB - rA = (xB - xA)*i + (yB - yA)*j + (zB - zA)*kを用いて、r_Aとr_Bから求めることができます。

例えば、点A(1 m, 0 m, -3 m)から点B(-2 m, 2 m, 3 m)への位置ベクトルを求める場合、先端Bの座標(-2 m, 2 m, 3 m)から先端Aの座標(1 m, 0 m, -3 m)を引くと、rAB={ -3*i + 2*j + 6*k} mとなります。

位置ベクトルは、空間内の別の点に対する 1 つの点の位置を決定するのに役立つ数学の基本的な概念です。それは...

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

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 unit vectors in the x, y, and z directions.

The vector r gives both the direction and the magnitude of the path from O to P. When a vector points from point A to point B, it may be written as rAB. Position vectors from the origin use one subscript only, such as rA and rB.

A vector from A to B can be found by subtracting the coordinates of A from the coordinates of B. The relation is rAB = rB - rA = (xB - xA)i + (yB - yA)j + (zB - zA)k. For example, if A is (1 m, 0 m, -3 m) and B is (-2 m, 2 m, 3 m), then rAB = {-3 i + 2 j + 6 k} m.

Using Position Vectors to Locate Points

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key terms and concepts

Learn how position vectors locate points in space, use i, j, k notation, and calculate a vector from one point to another with a clear example.

Using Position Vectors to Locate Points | 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 Position Vectors to Locate Points

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