Name: adding and multiplying vectors
Uploaded: 2023-09-22T11:49:19.000Z
Duration: 1 min 20 s
Channel: JoVE

Si considerino due vettori di forza in un sistema di coordinate bidimensionale.
Questi vettori possono essere aggiunti utilizzando la legge del parallelogramma, in cui le code dei vettori concorrenti sono unite insieme e le linee parallele a ciascun vettore sono tracciate per formare un parallelogramma.
Il vettore risultante può essere ottenuto disegnando una diagonale dalle code al punto di intersezione delle linee.
Allo stesso modo, i vettori possono essere aggiunti utilizzando la regola del triangolo, in cui due vettori sono disposti come due lati del triangolo uniti testa-coda e il terzo lato del triangolo rappresenta la grandezza e la direzione del vettore risultante.
La sottrazione vettoriale è simile all'aggiunta dell'inversione di un vettore a un altro. Di conseguenza, la differenza tra due vettori si ottiene utilizzando le stesse regole di addizione vettoriale.
Quando un vettore viene moltiplicato per uno scalare, la grandezza del vettore varia per la quantità scalare.
Se lo scalare è positivo, la direzione del vettore rimane la stessa e se è negativo, la direzione è invertita.

<ol>
	<li>Hibbeler, R.C. (2016). Ingegneria Meccanica \u2012 Statica e Dinamica. Hoboken, New Jersey: Pearson Prentice Hall. pagine 18 -19</li>
</ol>

operazioni vettoriali

I vettori sono grandezze fisiche che hanno sia un’intensità sia una direzione. Le operazioni tra vettori comprendono la somma, la sottrazione e il prodotto scalare.

Il prodotto di un vettore per un valore scalare viene chiamato prodotto scalare. Il risultato ottenuto è un nuovo vettore con intensità diversa. Se lo scalare è positivo, la direzione del vettore rimane la stessa; se invece è negativo, la direzione del vettore va invertita. Ad esempio, il prodotto della massa e della velocità dà come risultato il momento. In questo caso, la grandezza scalare massa viene moltiplicata per la grandezza vettoriale velocità, il che porta alla grandezza vettoriale quantità di moto. La direzione del vettore momento risultante è la stessa del vettore velocità.

I due vettori possono essere sommati geometricamente utilizzando la regola del parallelogramma per la somma vettoriale. Quando vengono sommati utilizzando questa regola, i due vettori sono rappresentati come lati adiacenti di un parallelogramma. Le code di entrambi i vettori vengono posizionate nell'origine e si tracciano le linee parallele a ciascun vettore per formare un parallelogramma. Si traccia poi la diagonale del parallelogramma che va dall'origine all'angolo opposto e che rappresenta il vettore risultante.. Il valore assoluto e la direzione del vettore risultante possono essere determinati misurando la lunghezza e gli angoli della diagonale del parallelogramma. La regola del parallelogramma può essere semplificata in uno scenario speciale in cui i due vettori sono collineari, ossia hanno la stessa linea d'azione. In questo caso, il vettore risultante può essere ottenuto tramite la somma algebrica o scalare dei due vettori.

Un altro metodo per sommare graficamente due vettori è il metodo punta-coda. Secondo questa regola, se due vettori consecutivi sono disposti l’uno con la coda sulla punta dell’altro, come due lati di un triangolo, il terzo lato del triangolo rappresenterà il valore assoluto e la direzione del vettore risultante. Per eseguire la somma vettoriale applicando questa regola occorre far partire il primo vettore nell’origine e il secondo con la coda sulla punta del primo. Il terzo lato del triangolo, ossia quello che congiunge l'origine con la punta del secondo vettore rappresenta il vettore risultante.

Il procedimento per la sottrazione dei vettori comporta sommare l'inverso (negativo) di un vettore a un altro vettore. Ciò significa che per ottenere la differenza tra i due vettori è possibile applicare le regole della somma vettoriale.

Operazioni vettoriali

I vettori sono grandezze fisiche che hanno sia grandezza che direzione. Le operazioni vettoriali includono l'addizione, la sottrazione e la moltiplicazione scalare.
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Education

Didattica

JoVE Core

Mechanical Engineering

Ingegneria Meccanica

Unit 1

Unità 1

Force Vectors

Vettori di forza

KEY TERMS AND CONCEPTS

TERMINI E CONCETTI CHIAVE

Vector Operations

adding and multiplying vectors

Vectors are quantities with both magnitude and direction, and their operations include addition, subtraction, and scalar multiplication. Scalar multiplication means a vector is multiplied by a scalar value, which creates a new vector with a different magnitude. If the scalar is positive, the direction stays the same. If the scalar is negative, the direction reverses.

A common example is momentum, which comes from multiplying mass by velocity. In this case, mass is the scalar quantity and velocity is the vector quantity. The result is momentum, and its direction matches the direction of the velocity vector.

Two vectors can be added geometrically with the parallelogram law. In this method, the two vectors are drawn as adjacent sides of a parallelogram with both tails at the origin. Lines are then drawn parallel to each vector to complete the shape.

The diagonal of the parallelogram, drawn from the origin to the opposite corner, shows the resultant vector. Its magnitude and direction can be found by measuring the diagonal's length and angles. When the two vectors are collinear, meaning they share the same line of action, the resultant can be found by algebraic or scalar addition.

Another way to add vectors is the triangle law of vector addition. Here, the vectors are arranged head to tail as two sides of a triangle. The third side of the triangle, from the origin to the end of the second vector, represents the resultant vector.

Vector subtraction uses the reverse, or negative, of one vector added to another vector. This lets the rules of vector addition be used to find the difference between two vectors.

Adding and Multiplying Vectors

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vector operations

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key terms and concepts

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Adding and Multiplying Vectors | 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.

mechanical engineering

Adding and Multiplying Vectors

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