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

Ein Positionsvektor ist ein fester Vektor, der die Position eines Punktes im Raum relativ zu einem anderen Punkt lokalisiert.
Stellen Sie sich eine Schrauböse vor, die an einer Wand befestigt ist, und ein daran befestigtes Kabel-OP. Richten Sie ein kartesisches Koordinatensystem mit Schrauböse am Ursprung und den Komponentenvektoren entlang der Achsen ein, um den Punkt P auf dem Kabel zu lokalisieren.
Ein Ortsvektor r, der vom Ursprung zum Punkt P gerichtet ist, kann in der kartesischen Vektorform ausgedrückt werden.
Für eine allgemeine Darstellung eines kartesischen Vektors betrachten Sie eine Halterung, die an einer Wand befestigt ist, und eine daran befestigte Schnur.
Seien nA und nB die Ortsvektoren, die sich vom Ursprung bis zu den Punkten A und B erstrecken.
Der resultierende Ortsvektor n von Punkt A nach B kann also unter Verwendung der Dreiecksregel und durch Ausdrücken von nA und nB in der kartesischen Vektorform erhalten werden.
i, j und k sind die Einheitsvektoren entlang der Komponenten des Ortsvektors n, die durch Subtraktion der Koordinaten von A von B gebildet werden.

<ol>
	<li>Russell C. Hibbeler. (2016). Technische Mechanik: Statik & Dynamik. Pearson. (Seite Nr. – 56,57)</li>
</ol>

positionierung vektoren

Ein Positionsvektor ist ein grundlegendes Konzept in der Mathematik, das hilft, die Position eines Punktes in Bezug auf einen anderen Punkt im Raum zu bestimmen. Es ist ein Vektor, der die Richtung und Entfernung zwischen zwei Punkten beschreibt. Positionsvektoren sind in den Bereichen Mathematik und Wissenschaft sehr nützlich, da sie räumliche Beziehungen darstellen und Berechnungen erleichtern. Zum Beispiel, wenn wir einen Punkt P(x, y, z) in Bezug auf den Koordinatenursprung O lokalisieren möchten, können wir einen Positionsvektor r definieren, der vom Ursprung O zum Punkt P verläuft. Diesen Vektor können wir in kartesischer Vektorform ausdrücken als: r = xi + yj + zk, wobei i, j und k die Einheitsvektoren in den Richtungen x, y und z sind. Der Positionsvektor r gibt uns die Richtung und Größe des Vektors vom Punkt O zum Punkt P. Betrachten wir einen Positionsvektor, der vom Punkt A zum Punkt B im Raum gerichtet ist. Dieser Vektor kann durch das Symbol r gekennzeichnet werden. Wir können diesen Vektor auch mit zwei Indizes bezeichnen, um die Punkte anzugeben, von und zu denen er gerichtet ist. Somit können wir r auch als rAB bezeichnen. Bitte beachten Sie, dass wenn die Positionsvektoren vom Koordinatenursprung ausgehen, sie nur mit einem Index bezeichnet werden, als rA und rB. Der Positionsvektor rAB kann aus rA und rB mit dem Ausdruck rAB = rB − rA = (xB − xA)i + (yB − yA)j + (zB − zA)k erhalten werden. Zum Beispiel, um einen Positionsvektor vom Punkt A zu B zu etablieren, werden die Koordinaten des Schwanzes A(1 m, 0 m, −3 m) von den Koordinaten des Kopfes B(−2 m, 2 m, 3 m) subtrahiert, was rAB = { −3i + 2j + 6k} m ergibt.

Positionierung Vektoren

Ein Positionsvektor ist ein grundlegendes Konzept in der Mathematik, das hilft, die Position eines Punktes in Bezug auf einen anderen Punkt im Raum zu bestimmen. Es ist ein ...

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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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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

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Using Position Vectors to Locate Points

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