Name: dot product and work in physics
Uploaded: 2023-04-30T10:49:28.000Z
Duration: 1 min 11 s
Channel: JoVE

K12 Standards

Florida Department of Education

High School

Physical Science

SC.912.P.12

SC.912.P.12.1

Scalar Product (Dot Product)

The dot or scalar product of two vectors is the product of their magnitudes and the cosine of the angle between them.
It is the projection of one vector in the direction of another multiplied by the magnitude of the other.
For instance, work is the dot product of force on an object and its displacement. When a force acts at an angle to the horizontal, only a part contributes to work.
If it acts horizontally, it contributes fully to work.
A vertical force cannot do any work in horizontal displacement.
The scalar product of two vectors is commutative and distributive.
The scalar product of the unit vector of one axis with another vanishes and with itself is one.
Using these, the dot product of two vectors is the sum of the products of corresponding components.
The angle between two vectors can be obtained using the dot product, and it is inverse cosine of the ratio of their dot product to the product of their magnitudes.

scalar product (dot product)

The scalar multiplication of two vectors is known as the scalar or dot product. As the name indicates, the scalar product of two vectors results in a number, that is, a scalar quantity. Scalar products are used to define work and energy relations. For example, the work that a force (a vector) performs on an object while causing its displacement (a vector) is defined as a scalar product of the force vector with the displacement vector.
The scalar product of two vectors is obtained by multiplying their magnitudes with the cosine of the angle between them. In the definition of the dot product, the direction of the angle between the two vectors does not matter and can be measured from either of the two vectors. The scalar product of orthogonal vectors vanishes. Moreover, the dot product of two parallel vectors is the product of their magnitudes, and likewise, the dot product of two antiparallel vectors is also the product of their magnitudes. The scalar product of a vector with itself is the square of its magnitude.
In the Cartesian coordinate system, scalar products of the unit vector of an axis with other unit vectors of axes always vanish because these unit vectors are orthogonal. The scalar multiplication of two vectors is commutative and obeys distributive law. The scalar product of two different unit vectors of axes is zero, and the scalar product of unit vectors with themselves is one. The scalar product of two vectors is used to find the angle between the vectors.
This text is adapted from <a href="https://openstax.org/books/university-physics-volume-1/pages/2-4-products-of-vectors">Openstax, University Physics Volume 1, Section 2.4: Products of Vectors.</a>

The scalar multiplication of two vectors is known as the scalar or dot product. As the name indicates, the scalar product of two vectors results in a ...

Education

JoVE Core

Physics

Unit 1

Vectors and Scalars

KEY TERMS AND CONCEPTS

dot product and work in physics

The dot product links two vectors to a single number. That number is called a scalar. In physics, this idea helps describe work and energy.

A common example is work done by a force on an object. Force is a vector, and displacement is also a vector. The work is defined as the scalar product of the force vector and the displacement vector.

To find the dot product, multiply the magnitudes of the two vectors by the cosine of the angle between them. The angle can be measured from either vector. The direction used does not change the result.

Perpendicular vectors, also called orthogonal vectors, have a dot product of zero. Parallel vectors have a dot product equal to the product of their magnitudes. Antiparallel vectors also give the product of their magnitudes.

A vector dotted with itself equals the square of its magnitude. In a Cartesian coordinate system, unit vectors on different axes have dot product zero because they are orthogonal. Unit vectors with themselves have dot product one.

The dot product is commutative and follows the distributive law. It is also useful for finding the angle between two vectors. These rules make scalar products an important tool for vector analysis in physics.

Dot Product and Work in Physics

introduction to scalars

Scalars in Everyday Physics

Scalar quantities are physical quantities that can be described completely by one number and a unit. A class period can last 50 min, a gas tank can hold 65 L, and the distance between two posts can be 100 m. In each case, the value is fully given with no direction needed. The word scalar is a synonym for number. Time, mass, distance, length, volume, temperature, and energy are all examples of scalar quantities. When two scalar quantities have the same units, they can be added or subtracted...

introduction to vectors

Vectors, Scalars, and Direction

Vectors and scalars describe physical quantities in different ways. Some quantities need both magnitude and direction to be fully specified. In those cases, the quantity is called a vector quantity. Examples include displacement, velocity, position, force, and torque. A quantity with only magnitude and no direction is a scalar quantity. The difference matters in everyday examples. If a rescue team knows only the distance to a distress signal, that is not enough to reach it quickly. They also...

vector components in the cartesian coordinate system

vector-components-in-the-cartesian-coordinate-system

Breaking Vectors into x and y Parts

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polar and cylindrical coordinates

Using Polar and Cylindrical Coordinates

Polar and cylindrical coordinates are useful ways to describe position when rotation or cylindrical shape is involved. Cartesian coordinates work well for displacement, velocity, and forces, but they can become awkward when an object turns or curves around an axis. In the polar coordinate system, a point in a plane is described by two values. The first is the radial coordinate, or the distance from the origin. The second is the angle made by the radial vector, usually measured from the...

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spherical coordinates

Spherical Coordinates for 3D Positioning

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vector algebra: graphical method

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Vectors can be added, subtracted, and multiplied by scalars. The vector sum of two or more vectors is called the resultant vector, or resultant. Geometric rules help find the resultant and its direction in a plane. For two vectors, the parallelogram rule shows how to build the sum. First, move one vector in parallel so both tails start at the same point. Then draw lines through the heads that are parallel to the opposite vector. The diagonal from the shared origin gives the resultant vector. <br...

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Vector Components for Adding Vectors

Vector components make it easier to add, subtract, and multiply vectors. The parallelogram rule and graphical methods can become hard to use when the math gets complex. One option is to draw vectors to scale, as in navigation, and estimate their lengths and directions from the graph. Another option is the method of components. In many situations, the size and direction of each vector are already known, and the goal is to find the resultant, or combined vector, of several vectors. A real-world...

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The dot product links two vectors to a single number. That number is called a scalar. In physics, this idea helps describe work and energy. A common example is work done by a force on an object. Force is a vector, and displacement is also a vector. The work is defined as the scalar product of the force vector and the displacement vector. To find the dot product, multiply the magnitudes of the two vectors by the cosine of the angle between them. The angle can be measured from either vector.

vector product (cross product)

Right-Hand Rule for Vector Multiplication

Vector multiplication gives a vector product, also called the cross product. Its magnitude equals the product of the two vectors times the sine of the angle between them. The result points perpendicular to both vectors. The direction of this perpendicular vector is found with the right-hand thumb rule. Because a plane has two possible perpendicular directions, the rule shows which one to choose. To use it, rotate the first vector about the perpendicular line until it lines up with the second...

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Triple Products in Vectors and Volume

Scalar and vector triple products extend the rules for multiplying two vectors to three vectors. A scalar product gives a scalar, while a vector product gives a vector. These same ideas help define two useful triple-product forms. The scalar triple product is the dot product of one vector with the cross product of two others. It produces a scalar quantity. When the vectors are rotated in a cyclic order, the result stays the same, so this form is associative. It also shows the projection of one...

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gradient and del operator

Gradient Direction on Scalar Fields

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divergence and curl

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second derivatives and laplace operator

Using Del Operators to Build Laplacians

The del operator can form important second-order expressions from gradients, divergence, and curl. These combinations matter in mathematics and physics. They help describe how scalar and vector functions change at a deeper level. For a scalar function, the curl of its gradient is one such expression. For a vector function, the divergence of a curl is another. The curl of a gradient and the divergence of a curl are always zero. The divergence of the gradient of a scalar function leads to the...

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line, surface, and volume integrals

Work, Circulation, and Flux Integrals

Work, circulation, flux, and volume integrals describe how scalar and vector fields add up over a path, surface, or region. For high school science, these ideas help connect math to motion, flow, and stored mass or energy. A line integral for a vector field adds the dot product of a vector function and a tiny displacement vector along a chosen path. If the path is closed, the result is a closed-line integral. The value around a closed path is called the circulation of the vector field. A...

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divergence and stokes' theorems

Divergence and Stokes Theorems in Vector Fields

Divergence and Stokes theorems help connect vector fields to the shapes they pass through. A vector field gives a value with direction at each point, such as electric flow or wind flow. These theorems extend Green's theorem to higher dimensions and also generalize the fundamental theorem of calculus. The divergence theorem focuses on the dot product of a vector field. It lets a difficult surface integral become a volume integral, and it can also work in reverse. With this theorem, the rate of...

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key terms and concepts

Learn how the dot product connects vectors to scalar quantities, including work, angles between vectors, and key rules for parallel and perpendicular vectors.

Dot Product and Work in Physics | JoVE K12

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SC.912.P.12.5

A. Motion can be measured and described qualitatively and quantitatively. Net forces create a change in motion. When objects travel at speeds comparable to the speed of light, Einstein's special theory of relativity applies.

B. Momentum is conserved under well-defined conditions. A change in momentum occurs when a net force is applied to an object over a time interval.

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E. Chemical reaction rates change with conditions under which they occur. Chemical equilibrium is a dynamic state in which forward and reverse processes occur at the same rates.

Dot Product and Work in Physics

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