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

The dot product of any two vectors is the product of the magnitude of the vectors and the cosine of the angle between them.
Consider an object being pulled by a vehicle along the ground by a rope. If the rope makes an angle theta with the horizontal axis, then the work done can be calculated using the dot product of the force applied to displace the object by a certain distance.
The dot product of any two vectors expressed in Cartesian form can be determined by multiplying their corresponding x, y, and z components and adding their products algebraically.
Suppose the angle theta is unknown, dot product can be used to determine the angle between the two vectors using the inverse cosine function.
The force component along the direction of the displacement can be determined by taking a dot product of the vector with a unit vector that defines its direction.
The dot product of vectors always follows the commutative law and the distributive law of addition and multiplication.

dot product

The dot product is an essential concept in mathematics and physics.
In engineering, the dot product of any two vectors is the product of the magnitudes of the vectors and the cosine of the angle between them. It is denoted by a dot symbol between the two vectors.
Consider a vehicle pulling an object along the ground using a rope. If the rope makes an angle with the horizontal axis, the work done can be calculated using the dot product of the force applied and the object's displacement.
The dot product of two vectors expressed in Cartesian form can be determined by multiplying their corresponding x, y, and z components and adding their products algebraically. This is given by the formula for the dot product of vectors A and B: 
<img alt="Dot product formula: A·B=(AxBx)+(AyBy)+(AzBz), mathematical equation, vector operations." src="/files/ftp_upload/14238/14238_Equation_1.svg" style="text-align: center; height: 15px;" data-altupdatedat="1754904545000" data-alt="Dot product equation, vector calculation formula, depicting components in a mathematical expression.">
If the angle between two vectors is unknown, the dot product can help determine the angle using the inverse cosine function. Suppose A and B are two vectors, and the angle between them can be determined by:
<img alt="Equation showing dot product formula for vector angle calculation, θ = cos⁻¹((A·B)/|A||B|)." src="/files/ftp_upload/14238/14238_Equation_2.svg" style="text-align: center; height: 30px;" data-altupdatedat="1754904545000" data-alt="Angle calculation using dot product formula, θ = cos⁻¹((A·B)/(AB)), mathematical concept.">
The dot product follows the commutative and distributive laws of addition and multiplication. The dot product is a fundamental operation used in many areas of physics and engineering. It can calculate work done, find the angle between vectors, and determine force components along a specific direction. With its many applications, the dot product is an important concept in vector algebra.

Dot Product

The dot product is an essential concept in mathematics and physics.
In engineering, the dot product of any two vectors is the product of the magnitudes of ...

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Mechanical Engineering

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Force Vectors

KEY TERMS AND CONCEPTS

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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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...

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The dot product helps solve vector questions by combining vector size and the cosine of the angle between them. It is useful when the angle is measured anti-clockwise and when a problem asks for a vector relationship in component form. A good first step is to identify the question being asked. Then list the given vectors and write them in Cartesian form, also called component form. This makes the data easier to use in a calculation. Next, decide whether the dot product is the right operation.

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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

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Using Dot Product to Find Work and Angle

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