Name: right-hand rule for vector multiplication
Uploaded: 2023-04-30T10:49:35.000Z
Duration: 1 min 17 s
Channel: JoVE

K12 Standards

Florida Department of Education

High School

Physical Science

SC.912.P.12

SC.912.P.12.1

Vector Product (Cross Product)

The cross or vector product of two vectors is the product of their magnitudes and the sine of the angle. It is directed perpendicular to the vectors&#39;&#160;plane.
Geometrically, it is the area of the parallelogram spanned by the vectors.
Using the right-hand rule, if the index finger and the middle fingers of the right-hand point the first and the second vectors, respectively,&#160; the thumb finger points to their cross product.
The cross product of the second vector with the first has the opposite direction. Hence, order matters for cross products.
The measure of a door&#39;s rotation results from a torque which is the cross product of the radial distance between the hinge and the point of application of force with the applied force.
The door does not rotate if a force acts along or opposite the direction of radial distance.
For the unit vector of an axis, its cross product with itself is zero. But with another, is the third one in cyclic-order.
The components of the cross product of vectors A and B are AyBz minus AzBy, AzBx minus AxBz, and AxBy minus AyBx.

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 the angle between both the vectors and the direction perpendicular to both the individual vectors. As there are always two directions perpendicular to a given plane, one on each side, the direction of the vector product is governed by the right-hand thumb rule.
Consider the cross product of two vectors. Imagine rotating the first vector about the perpendicular line until it is aligned with the second, choosing the smaller of the two possible angles between the two vectors. The fingers of the right hand are curled around the perpendicular line so that the fingertips point in the direction of rotation; the thumb then points in the direction of the cross product. Similarly, the direction of the cross product of the second vector with the first is determined by rotating the second vector into the first vector, and it is opposite to the cross product of the first vector with the second. This implies that these two are antiparallel to each other. The cross product of two vectors is anti-commutative, which means that the vector product reverses the sign when the order of multiplication is reversed. The vector product of two parallel or antiparallel vectors is always zero. In particular, the vector product of any vector with itself is zero.
Let us consider a few cases where the vector product is applied. The mechanical advantage that a wrench provides depends on the magnitude of the applied force, on its direction with respect to the wrench handle, and on how far from the nut this force is applied. The physical vector quantity that makes the nut turn is called torque, and it is the vector product of the applied force vector with the position vector.&#160; Another example is the case of a particle moving in a magnetic field that experiences a magnetic force. The magnetic field, magnetic force, and velocity are vector quantities. The force vector is proportional to the vector product of the velocity vector with the magnetic field vector.
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>

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 ...

Education

JoVE Core

Physics

Unit 1

Vectors and Scalars

KEY TERMS AND CONCEPTS

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 vector. Use the smaller angle between the two vectors. Curl the fingers of your right hand in the direction of rotation, and the thumb points along the cross product.

The order of the vectors matters. If the second vector is rotated into the first, the cross product points in the opposite direction. That means the two results are antiparallel. The cross product is anti-commutative, so reversing the order changes the sign.

When two vectors are parallel or antiparallel, their vector product is zero. The product of any vector with itself is also zero. These rules help explain real uses of the cross product.

One example is torque from a wrench. The turning effect depends on the applied force, the direction of that force relative to the wrench handle, and the distance from the nut. Torque is the vector product of the applied force vector and the position vector. Another example is magnetic force on a moving particle. In that case, the magnetic field, magnetic force, and velocity are all vectors, and the force is proportional to the vector product of velocity and magnetic field.

Right-Hand Rule for Vector Multiplication

right-hand-rule-for-vector-multiplication

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...

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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...

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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...

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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 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

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

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...

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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.

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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 right-hand rule gives the direction of the cross product, why order changes the sign, and how it explains torque and magnetic force.

Right-Hand Rule for Vector Multiplication | JoVE K12

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Right-Hand Rule for Vector Multiplication

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