Name: triple products in vectors and volume
Uploaded: 2023-05-16T08:18:40.000Z
Duration: 1 min 6 s
Channel: JoVE

K12 Standards

Florida Department of Education

High School

Physical Science

SC.912.P.12

SC.912.P.12.1

Scalar and Vector Triple Products

The scalar triple product is the dot product of a vector with the cross product of two vectors. It results in a scalar quantity.
Here, the magnitude of the cross product of two vectors gives the area of the parallelogram formed by these vectors. The resultant vector is perpendicular to the parallelogram plane.
The projection of the third vector in the direction of the resultant of the cross product gives the height of the parallelepiped.
It follows that the resultant of the scalar triple product denotes the volume of the parallelopiped formed by these vectors.
The volume remains unaltered if the vectors are rotated in cyclic order in the scalar triple product.
The Vector triple product is the cross product of a vector with the resultant of the cross product of two vectors and results in a vector quantity.
Here, if the vectors are rotated in cyclic order, the resultant vector is entirely new. So, the vector triple product is not associative.

scalar and vector triple products

Two vectors can be multiplied using a scalar product or a vector product. The resultant of a scalar product is scalar, while with vector products, the resultant is a vector. These rules of the scalar or vector product between two vectors can be applied to multiple vectors to obtain meaningful combinations. The scalar triple product is the dot product of a vector with the cross product of two vectors.
The scalar triple product is the dot product of a vector with the cross product of two vectors. As before, the scalar triple product results in a scalar quantity. The scalar triple product of three vectors can be expressed as follows:
<img alt="Vector triple product identity equations; math concept; cross and dot product application." src="/files/ftp_upload/14688/14688_Equation_1.svg" style="text-align: center; height: 20px;" data-altupdatedat="1761830439000" data-alt="Equation1" />
On the cyclic rotation of vectors, the result of the scalar triple product remains the same, meaning, it is associative. The scalar triple product is the projection of a vector onto the resultant of the cross product of two vectors and represents the volume defined by these three vectors.
On the other hand, the vector triple product is the cross product of a vector with the cross product of two other vectors, and it results in a vector quantity. The vector triple product of three vectors can be expressed as follows:
<img alt="Vector triple product identity equation: \( \vec{A} \times (\vec{B} \times \vec{C}) = \vec{B} (\vec{A} \cdot \vec{C}) - \vec{C} (\vec{A} \cdot \vec{B}) \) formula" src="/files/ftp_upload/14688/14688_Equation_2.svg" style="text-align: center; height: 20px;" data-altupdatedat="1761830439000" data-alt="Equation2" />
Here, the cyclic rotation of vectors results in a new vector. The vector triple product is not associative.

Two vectors can be multiplied using a scalar product or a vector product. The resultant of a scalar product is scalar, while with vector products, the ...

/files/media/science-education/science-education-thumbs/14688.jpg

Education

JoVE Core

Physics

Unit 1

Vectors and Scalars

KEY TERMS AND CONCEPTS

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 vector onto the cross product of the other two and represents the volume enclosed by the three vectors.

The vector triple product is the cross product of one vector with the cross product of two other vectors. It produces a vector quantity. A cyclic rotation of the vectors gives a new vector, so this form is not associative.

Triple Products in Vectors and Volume

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

Vectors in a Cartesian coordinate system are often split into x and y parts. This makes it easier to describe direction and size on a rectangular plane. It also matches the way people give directions in everyday life, such as east, west, north, south, and an angle of travel. In a plane, a point is written as an ordered pair, (x, y). A vector is described in a similar way with its vector coordinates. The x-coordinate of the vector is its x-component, and the y-coordinate is its y-component.

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...

/files/media/science-education/science-education-thumbs/12611.jpg

spherical coordinates

Spherical Coordinates for 3D Positioning

Spherical coordinates describe position in three-dimensional space using radius, polar angle, and azimuthal angle. They are especially useful for systems with spherical symmetry, where the shape matches a sphere. Compared with Cartesian coordinates, spherical coordinates can make calculations simpler. To describe a sphere, Cartesian coordinates need three values. Spherical coordinates can use just one parameter, the radius. That is why they are common in science and engineering problems such...

/files/media/science-education/science-education-thumbs/14691.jpg

vector algebra: graphical method

Adding and Subtracting Vectors Graphically

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...

vector algebra: method of components

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...

/files/media/science-education/science-education-thumbs/12613.jpg

scalar product (dot product)

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.

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...

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...

gradient and del operator

Gradient Direction on Scalar Fields

The gradient helps describe how a scalar field changes in space. It gives both the size and the direction of the greatest rate of change. In physics and mathematics, this makes it a key tool for studying functions and fields. A simple mountain example shows why direction matters. A slope does not mean much unless it is measured in a specific direction. That is why the directional derivative is used. It is a vector that gives the slope and the direction of change. The gradient of a scalar...

/files/media/science-education/science-education-thumbs/14689.jpg

divergence and curl

Vector Fields: Sources, Sinks, and Rotation

Vector fields can spread outward, pull inward, or rotate around a point. Divergence describes the net outward flow of flux from a small volume through a closed surface as the volume gets very small. In simple terms, it shows how much a vector field spreads out from a point. When the flux moves outward, the divergence is positive. A point with positive divergence is often called a source. When the field contracts toward a point, the divergence is negative. That point is often called a sink.

/files/media/science-education/science-education-thumbs/14690.jpg

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...

/files/media/science-education/science-education-thumbs/14692.jpg

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...

/files/media/science-education/science-education-thumbs/14693.jpg

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...

/files/media/science-education/science-education-thumbs/14694.jpg

key terms and concepts

Learn how scalar triple products give a scalar volume measure and how vector triple products produce a new vector through cross products and dot products.

Triple Products in Vectors and Volume | JoVE K12

physics

JoVE Core Physics introduces the theory and application of fundamental physics topics, including kinematics, dynamics, fluid mechanics, statics, waves, and acoustics, with concise and easy-to-understand animated videos. Additionally, the scientist-in-action videos demonstrate the application of related concepts in classical and original research experiments performed in today's laboratories worldwide.

Introduction to Scalars

Many familiar physical quantities can be specified completely by giving a single number and the appropriate unit. For example, &#34;a class period lasts ...

Introduction to Vectors

To define some physical quantities, there is a need to specify both magnitude as well as direction. For example, when the U.S. Coast Guard dispatches a ...

Vector Components in the Cartesian Coordinate System

Vectors are usually described in terms of their components in a coordinate system. Even in everyday life, we naturally invoke the concept of orthogonal ...

Vector Algebra: Graphical Method

Vectors can be multiplied by scalars, added to other vectors, or subtracted from other vectors. The vector sum of two (or more) vectors is called the ...

Vector Algebra: Method of Components

It is cumbersome to find the magnitudes of vectors using the parallelogram rule or using the graphical method to perform mathematical operations like ...

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 ...

Vector Product (Cross Product)

Vector multiplication of two vectors yields a vector product, with the magnitude equal to the product of the individual vectors multiplied by the sine of ...

Gradient and Del Operator

In mathematics and physics, the gradient and del operator are fundamental concepts used to describe the behavior of functions and fields in space. The ...

Divergence and Curl

The divergence of a vector field at a point is the net outward flow of the flux out of a small volume through a closed surface enclosing the volume, as ...

Second Derivatives and Laplace Operator

The first order operators using the del operator include the gradient, divergence and curl. Certain combinations of first order operators on a scalar or ...

Line, Surface, and Volume Integrals

A line integral for a vector field is defined as the integral of the dot product of a vector function with an infinitesimal displacement vector along a ...

Distinguish between scalar and vector quantities and assess which should be used to describe an event.

Position and Displacement

The position of an object defines its location relative to a convenient frame of reference at any particular time. A frame of reference is an arbitrary ...

Average Velocity

To calculate the other physical quantities in kinematics, we must introduce the time variable. The time variable allows us not only to state the position ...

Instantaneous Velocity - I

The average velocity during a time interval cannot tell us how fast or in what direction a particle is moving at any given time during the interval. To ...

Instantaneous Velocity - II

Instantaneous velocity is the quantity that measures how fast an object is moving along its path. In other words, the instantaneous velocity of an object ...

Average Acceleration

The importance of understanding acceleration spans our day-to-day experiences, as well as the vast reaches of outer space and the tiny world of subatomic ...

Instantaneous Acceleration

Acceleration is in the direction of the change in velocity, but it is not always in the direction of motion. When an object slows down, its acceleration ...

Kinematic Equations - I

When an object moves with constant acceleration, the velocity of the object changes at a constant rate throughout the motion. The kinematic equations of ...

Kinematic Equations - II

The second kinematic equation expresses the final position of an object in terms of its initial position, the distance traveled with the initial constant ...

Kinematic Equations - III

The first two kinematic equations have time as a variable, but the third kinematic equation is independent of time. This equation expresses final velocity ...

Kinematic Equations: Problem Solving

When analyzing one-dimensional motion with constant acceleration, the problem-solving strategy involves identifying the known quantities and choosing the ...

Free-falling Bodies: Introduction

All objects, neglecting air resistance, fall with the same acceleration towards the Earth&#39;s center due to the force exerted by the Earth&#39;s ...

Free-falling Bodies: Example

An object falling without any air resistance under the influence of gravitational force is said to be in free-fall. For free-falling bodies, the ...

velocity-and-position-by-graphical-method

Velocity and Position by Graphical Method

Velocity and position can be calculated from the known function of acceleration as a function of time. The total area under the acceleration-time graph ...

Non-uniform Circular Motion

In&#160;uniform circular motion, the particle executing circular motion has a constant speed, and the circle is at a fixed radius. However, not all ...

Velocity and Position by Integral Method

If acceleration as a function of time is known, then velocity and position functions can be derived using integral calculus. For constant acceleration, ...

Relative Velocity in One Dimension

The understanding of the concept of reference frames is essential to discuss relative motion in one or more dimensions. When we say that an object has a ...

Position and Displacement Vectors

To describe the motion of an object, one should first be able to describe its position (where it is at any particular time). More precisely, the position ...

Relative Velocity in Two Dimensions

Relative velocity is the velocity of an object as observed from a particular reference frame, or the velocity of one reference frame with respect to ...

average-and-instantaneous-velocity-vectors

Average and Instantaneous Velocity Vectors

To calculate other physical quantities in kinematics, the time variable must be introduced. The time variable not only allows us to state where an object ...

Acceleration Vectors

In everyday conversation, accelerating means speeding up. Acceleration is a vector in the same direction as the change in velocity, &#916;v, therefore the ...

Direction of Acceleration Vectors

Acceleration occurs when velocity changes in magnitude (an increase or decrease in speed), direction, or both. Although acceleration is in the direction ...

Projectile Motion

An object thrown in the air follows a parabolic path under the influence of Earth&#39;s gravitational force. The motion of such an object is called ...

/files/media/science-education/science-education-thumbs/12634.jpg

Projectile Motion: Equations

Projectile motion is commonly observed in our day-to-day life. For example, a basketball thrown by a player, an arrow shot from a bow, and kids jumping ...

Projectile Motion: Example

The theory of projectile motion is very useful for players of several sports to improve their performance. For example, a javelin thrower needs to throw ...

Uniform Circular Motion

Uniform circular motion is a specific type of motion in which an object travels in a circle with a constant speed. For example, any point on a propeller ...

Analyze the motion of an object in terms of its position, velocity, and acceleration (with respect to a frame of reference) as functions of time.

SC.912.P.12.2

Newton's First Law: Introduction

Motion draws our attention. Motion itself can be beautiful, causing us to marvel at the forces needed to create spectacular sights, such as that of a ...

Newton's First Law: Application

Experience suggests that an object at rest remains at rest if left alone, and that an object in motion tends to slow down and stop unless some effort is ...

Internal and External Forces

Newton&#39;s first law states that a net external force causes a change in motion. External forces act on an object or system, originating outside of the ...

Newton's Second Law

Newton&#39;s second law is closely related to his first law of motion. It mathematically gives the cause-and-effect relationship between force and changes ...

Newton's Third Law: Introduction

Whenever one body exerts a force on a second body, the first body experiences a force equal in magnitude and opposite in direction, to the force that it ...

Newton's Third Law: Examples

Newton&#39;s third law states that every action has an equal and opposite reaction. Consider a swimmer pushing off the side of a pool. They push against ...

Inertial Frames of Reference

Newton&#8217;s first law is usually considered to be a statement about reference frames. It provides a method for identifying a special type of reference ...

Non-inertial Frames of Reference

A reference frame accelerating or decelerating relative to an inertial frame is a non-inertial frame. To help understand this, consider what taking off in ...

first-law-particles-in-one-dimensional-equilibrium

First Law: Particles in One-dimensional Equilibrium

Newton&#39;s first law of motion states that a body at rest remains at rest, or if in motion, remains in motion at constant velocity, unless acted on by a ...

first-law-particles-in-two-dimensional-equilibrium

First Law: Particles in Two-dimensional Equilibrium

Recall that a particle in equilibrium is one for which the external forces are balanced. Static equilibrium involves objects at rest, and dynamic ...

Second Law: Motion under Same Force

Newton&#39;s laws can be applied to bodies at rest and bodies in motion. Newton&#39;s first law is applied to bodies in equilibrium, whereas the second ...

second-law-motion-under-same-acceleration

Second Law: Motion under Same Acceleration

Newton&#39;s second law of motion applies to bodies moving under the same acceleration. For example, when a baggage tractor pulls luggage carts, each cart ...

Interpret and apply Newton's three laws of motion.

SC.912.P.12.3

Gravitation

In the years before Newton, a general belief prevailed that different laws governed objects in the sky than objects on Earth. When Kepler wrote down the ...

/files/media/science-education/science-education-thumbs/14716.jpg

Newton's Law of Gravitation

Our everyday observation tells us that all objects close to the Earth naturally tend to fall to the ground. Early philosophers assumed that this downward ...

gravitation-between-spherically-symmetric-masses

Gravitation Between Spherically Symmetric Masses

The gravitational potential energy between two spherically symmetric bodies can be calculated from the masses and the distance between the bodies, ...

/files/media/science-education/science-education-thumbs/14717.jpg

Gravity between Spherical Bodies

Newton&#39;s law of gravitation describes the gravitational force between any two point masses. However, for extended spherical objects like the Earth, ...

Describe how the gravitational force between two objects depends on their masses and the distance between them.

SC.912.P.12.4

Linear Momentum

The term momentum is used in various ways in everyday language, most of which are consistent with the precise scientific definition. Generally, momentum ...

Force and Momentum

Force and momentum are intimately related. Force acting over time can change momentum, and Newton&#39;s second law of motion can be stated in its most ...

Apply the law of conservation of linear momentum to interactions, such as collisions between objects.

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.

C. The Law of Universal Gravitation states that gravitational forces act on all objects irrespective of their size and position.

D. Gases consist of great numbers of molecules moving in all directions. The behavior of gases can be modeled by the kinetic molecular theory.

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.

Triple Products in Vectors and Volume

Description

Transcript

Tags

More videos by Standards