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

O produto escalar de quaisquer dois vetores é o produto da magnitude dos vetores e do cosseno do ângulo entre eles.
Considere um objeto sendo puxado por um veículo ao longo do chão por uma corda. Se a corda fizer um ângulo com o eixo horizontal, o trabalho realizado pode ser calculado usando o produto escalar da força aplicada para deslocar o objeto a uma certa distância.
O produto escalar de quaisquer dois vetores expressos na forma cartesiana pode ser determinado multiplicando seus componentes x, y e z correspondentes e adicionando seus produtos algebricamente.
Suponha que o ângulo seja desconhecido, o produto escalar pode ser usado para determinar o ângulo entre os dois vetores usando a função cosseno inversa.
A componente de força ao longo da direção do deslocamento pode ser determinada tomando um produto escalar do vetor com um vetor unitário que define sua direção.
O produto escalar dos vetores sempre segue a lei comutativa e a lei distributiva de adição e multiplicação.

<ol>
	<li>Hibbeler, R.C. (2016). Engenharia Mecânica \u2012 Estática e Dinâmica. Hoboken, Nova Jersey: Pearson Prentice Hall. pág. 70</li>
</ol>

produto escalar

O produto escalar é um conceito essencial em matemática e física.

Na engenharia, o produto escalar de dois vetores é o produto das magnitudes dos vetores pelo cosseno do ângulo entre eles. É representado por um símbolo de ponto entre os dois vetores.

Considere um veículo puxando um objeto ao longo do solo usando uma corda. Se a corda fizer um ângulo com o eixo horizontal, o trabalho realizado pode ser calculado usando o produto escalar da força aplicada e o deslocamento do objeto.

O produto escalar de dois vetores expressos na forma cartesiana pode ser determinado multiplicando seus respectivos componentes x, y e z e adicionando seus produtos de forma algébrica. Isso é dado pela fórmula para o produto escalar dos vetores A e B:

<img alt="Equation 1" src="/files/ftp_upload/14238/14238_Equation_1.svg" style="text-align: center; height: 15px;" />

Se o ângulo entre dois vetores for desconhecido, o produto escalar pode ajudar a determinar o ângulo usando a função cosseno inverso. Suponha que A e B sejam dois vetores e o ângulo entre eles possa ser determinado por:

<img alt="Equation 1" src="/files/ftp_upload/14238/14238_Equation_2.svg" style="text-align: center; height: 30px;" />

O produto escalar segue as leis comutativa e distributiva da adição e multiplicação. O produto escalar é uma operação fundamental usada em muitas áreas da física e engenharia. Ele pode calcular o trabalho realizado, encontrar o ângulo entre vetores e determinar componentes de força ao longo de uma direção específica. Com suas muitas aplicações, o produto escalar é um conceito importante na álgebra vetorial.

Produto Escalar

O produto escalar é um conceito essencial em matemática e física.
Na engenharia, o produto escalar de quaisquer dois vetores é o produto das magnitudes de ...

Education

JoVE Core

Mechanical Engineering

Engenharia Mecânica

Unit 1

Unidade 1

Force Vectors

Vetores de Força

KEY TERMS AND CONCEPTS

TERMOS-CHAVE E CONCEITOS

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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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

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Using Dot Product to Find Work and Angle

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