Name: vectors, scalars, and direction
Uploaded: 2023-04-30T10:48:50.000Z
Duration: 1 min 21 s
Channel: JoVE

K12 Standards

ohios-learning-standards-and-model-curriculum

Ohio’s Learning Standards and Model Curriculum

9-12

Physical Science

Forces And Motion

Motion

Introduction to Vectors

Vector quantities are physical quantities having both magnitude and direction.
Magnitude is the size or amount of a quantity, and the direction is the orientation in space.
For example, to open a door, a certain amount of force is applied in a particular direction. Hence, force is a vector quantity.
The weight of any object, which is the force exerted on a body due to gravity, is a vector quantity.
Vectors are represented as boldfaced italicized letters with an arrow above them. The magnitude is indicated with the same vector symbol between vertical bars or using italicized letters.
Geometrically, vectors are shown using a line with an arrow pointing its direction and length indicating its magnitude.
The displacement of an object from points A to B is shown with an arrow originating at A and ending at B and the vector is denoted as vector DAB.

Two vectors are equal if they have the same magnitude and direction.
The negative of a vector is obtained by reversing its direction.
If two vectors point in the same direction, they are parallel. If they point in opposite directions, they are antiparallel.

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 ship or a helicopter for a rescue mission, the rescue team needs to know not only the distance to the distress signal, but also the direction from which the signal is coming, so that they can get to it as quickly as possible. Physical quantities specified completely with a number of units (magnitude) and a direction are called vector quantities. Examples of vector quantities include displacement, velocity, position, force, and torque. In the language of mathematics, physical vector quantities are represented by mathematical objects called vectors. We denote vector quantities by an italicized letter with an arrow above it. A quantity with only a magnitude and no direction is a scalar quantity.&#160;Therefore, weight is a vector quantity, whereas mass is a scalar quantity. Similarly, speed is a scalar quantity, whereas velocity is a vector quantity.
Suppose you tell a friend on a camping trip that you have discovered a terrific fishing hole 6 km from your tent. It is unlikely your friend would be able to find the hole easily unless you also provide the direction in which it can be found from your campsite. You may say, for example, &#34;Walk about 6 km northeast from my tent.&#34;&#160;The key concept here is that you must give two pieces of information, namely the distance or magnitude (6 km) and the direction (northeast). A change in position, such as from the tent to the fishing hole in this example, is known as displacement. This is an example of a vector quantity. Geometrically, vectors are represented as arrows; their length (which is a positive number) gives the magnitude, indicated by placing the absolute value notation around the symbol that denotes the vector. The point of the arrow gives the direction, and the arrowhead marks the end point of the vector.
Two vectors are equal if and only if they have equal magnitudes and the same direction. Two vectors that have identical directions are said to be parallel vectors&#8212;meaning they are parallel to each other. However, if a vector points in the opposite direction, which is exactly 180&#176; to the first vector, they are said to be antiparallel. Two vectors with directions perpendicular to each other are said to be orthogonal vectors.
This text is adapted from <a href="https://openstax.org/books/university-physics-volume-1/pages/2-1-scalars-and-vectors">Openstax, University Physics Volume 1, Section 2.1: Scalars and Vectors.</a>

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 ...

Education

JoVE Core

Physics

Unit 1

Vectors and Scalars

KEY TERMS AND CONCEPTS

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 need the direction. A camping trip example shows the same idea. If a fishing hole is 6 km from a tent, a friend needs both the distance and the direction, such as northeast, to find it. This change in position is called displacement, and it is a vector quantity.

Vectors are represented in mathematics by arrows. The length of the arrow shows the magnitude, which is a positive number. The arrow points in the direction of the vector, and the arrowhead marks the end point. Vector quantities are often written with an arrow above an italicized letter. The magnitude of a vector is shown with absolute value notation around its symbol.

Some common examples help compare the two types of quantities. Weight is a vector quantity, while mass is a scalar quantity. Speed is a scalar quantity, while velocity is a vector quantity. Two vectors are equal only when they have the same magnitude and the same direction. Vectors with the same direction are parallel. Vectors in exactly opposite directions, 180° apart, are antiparallel. Vectors at right angles to each other are orthogonal.

Vectors, Scalars, and Direction

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...

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 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

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 vectors differ from scalars using displacement, velocity, and force. See how arrows show magnitude and direction in high school physics.

Vectors, Scalars, and Direction | 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.

Forces, Momentum And Motion

Forces in two dimensions

Arizona Science Standards

High School

HS+Phy.P3

HS+Phy.P3U1.3

Alabama Course of Study Science

motion-and-stability-forces-and-interactions

Motion and Stability: Forces and Interactions

9-12.P.1.1

Florida Department of Education

SC.912.P.12

SC.912.P.12.1

Texas Essential Knowledge and Skills

PHYS.5

PHYS.5.B

Georgia Standards of Excellence

SP1.

Scalar and Vector Quantities

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 ...

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 ...

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 ...

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 ...

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 ...

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 ...

• Introduction to one-dimensional vectors 
• Displacement, velocity (constant, average and instantaneous) and acceleration 
• Interpreting position vs. time and velocity vs. time graphs

Content elaboration: The motion of an object depends on the observer's frame of reference and is described in terms of distance, position, displacement, speed, velocity, acceleration and time. Position, displacement, velocity and acceleration are all vector properties (magnitude and direction). All motion is relative to whatever frame of reference is chosen for there is no motionless frame from which to judge all motion. The relative nature of motion will be addressed conceptually, not mathematically. Non-inertial reference frames are excluded. Motion diagrams can be drawn and interpreted to represent the position and velocity of an object. Showing acceleration on motion diagrams is reserved for Physics. The displacement or change in position of an object is a vector quantity that can be calculated by subtracting the initial position from the final position $(\Delta x=x_{f}-x_{i})$. Displacement can be positive or negative depending upon the direction of motion. Displacement is not always equal to the distance travelled. Examples should be given where the distance is not the same as the displacement. Velocity is a vector quantity that represents the rate at which position changes. Average velocity can be calculated by dividing displacement (change in position) by the elapsed time $(v_{avg}=(x_{f}-x_{i})/(t_{f}-t_{i}))$. Velocity may be positive or negative depending upon the direction of motion. Velocity should be distinguished from speed, which is always positive. Provide examples of when the average speed is not the same as the average velocity. Objects that move with constant velocity have the same displacement for each successive time interval. While speeding up or slowing down and/or changing direction, the velocity of an object changes continuously, from instant to instant. The speed of an object at any instant (clock reading) is called instantaneous speed. Acceleration is a vector quantity that represents the rate at which velocity changes. Average acceleration can be calculated by dividing the change in velocity by elapsed time $(a_{avg}=(v_{f}-v_{i})/(t_{f}-t_{i}))$. At this grade level, it should be noted that acceleration can be positive or negative, but specifics about what kind of motions produce positive or negative accelerations will be addressed in Physics. Deceleration is an ambiguous term that should only be used when an object is slowing down. Care should be given to ensure students do not associate negative acceleration with only deceleration. [cite: 352, 357] Objects with negative acceleration could be increasing their speed. Objects that have no acceleration can either be standing still or be moving with constant velocity (speed and direction). Constant acceleration occurs when the change in an object's instantaneous velocity is the same for equal successive time intervals. Motion can be represented by position vs. time and velocity vs. time graphs. Specifics about the speed, direction and change in motion can be determined by interpreting such graphs. For this course, graphs will be limited to positive x-values and show only uniform motion involving segments of constant velocity or constant acceleration. Motion can be investigated by collecting and analyzing data in the laboratory and should include constant velocity as well as constant acceleration. Technology can enhance motion exploration and investigation through video analysis, the use of motion detectors and graphing data for analysis. Objects that move with constant velocity and have no acceleration form a straight line (not necessarily horizontal) on a position vs. time graph. Objects that are at rest will form a horizontal line on a position vs. time graph. Objects that are accelerating will show a curved line on a position vs. time graph. Velocity can be calculated by determining the slope of a position vs. time graph. Positive slopes on position vs. time graphs indicate motion in a positive direction. time graphs indicate motion in a negative direction. While it is important that students can construct graphs by hand, computer graphing programs or graphing calculators can also be used so more time can be spent on graph interpretation and analysis. Constant acceleration is represented by a straight line (not necessarily horizontal) on a velocity vs. time graph. Objects that have no acceleration (at rest or moving at a constant velocity) will have a horizontal line for a velocity vs. time graph. Average acceleration can be determined from the slope of a velocity vs. time graph. The details about motion graphs should not be taught as rules to memorize, but rather as generalizations that can be developed from interpreting the graphs.

Force

Forces affect every moment of our life. Our bodies are held to the Earth by force, and they are held together by the forces of charged particles. When we ...

Types of Forces

In most situations, forces can be grouped into two categories: contact forces and field forces.&#160; Contact forces occur as a result of direct physical ...

Drawing Free-body Diagrams: Rules

The first step in describing and analyzing most phenomena in physics involves the careful drawing of a free-body diagram. Free-body diagrams are useful in ...

Free Body Diagrams: Examples

Solving problems that involve forces is easy using free-body diagrams. A free-body diagram is a sketch showing all the external forces that are acting on ...

Frictional Force

When a body is in motion, it encounters resistance because the body interacts with its surroundings. This resistance is known as friction, a common yet ...

Static and Kinetic Frictional Force

One of the simpler characteristics of sliding friction is that it is parallel to the contact surfaces between systems, and is always in a direction that ...

Drag Force and Terminal Speed

An interesting force in everyday life is the force of drag on an object when it is moving in a fluid. Like friction, the drag force always opposes the ...

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Tension

Tension is a force along the length of a medium, in particular, a force carried by a flexible medium, such as a rope or cable. The word &#34;tension&#34; ...

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Mass and Weight

Mass and weight are often used interchangeably in everyday conversation. For example,&#160; medical records often show our weight in kilograms, but never ...

Coulomb's Law

Experiments with electric charges have shown that if two objects each have an electric charge, they exert an electric force on each other. The magnitude ...

Magnetic Force

In addition to the electric forces between electric charges, moving electric charges exert magnetic forces on each other. A magnetic field is created by a ...

the-principle-of-superposition-and-the-gravitational-field

The Principle of Superposition and the Gravitational Field

The principle of superposition applies to gravitational forces of objects that are sufficiently far apart. It states that the net gravitational force on a ...

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Electric Field

Consider two point charges, each exerting Coulomb force on the other. It is possible to describe the Coulomb interaction via an intermediate step by ...

Magnetic Fields

A moving charge or a current creates a magnetic field in the surrounding space, in addition to its electric field. The magnetic field exerts a force on ...

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 ...

Frequency of Spring-Mass System

One interesting characteristic of the simple harmonic motion (SHM) of an object attached to a spring is that the angular frequency, and the period and ...

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• Force diagrams 
• Types of forces (gravity, friction, normal, tension) 
• Field model for forces at a distance

Content elaboration: Force is a vector quantity, having both magnitude and direction. Force diagrams are useful tools for visualizing and analyzing the forces acting on objects. The (SI) unit of force is a newton. One newton of net force will cause a 1 kg object to experience an acceleration of $1~m/s^{2}$. A newton can also be represented as $kg\cdot m/s^{2}$. The opportunity to measure force in the lab is provided (e.g., with a spring scale or a force probe). Normal forces and tension forces are introduced conceptually at this level. These forces and other forces introduced in prior grades (friction, drag, gravitational, electric and magnetic) can be used as examples of forces that affect motion. In this course, only forces in one dimension (positive and negative) will be addressed. The net force can be determined by one-dimensional vector addition. Gravitational force (weight) can be calculated from mass, but all other forces will only be quantified from force diagrams. Friction is a force that opposes motion. Kinetic friction (e.g., sliding, rolling), drag and static friction can be addressed conceptually. More quantitative study of friction forces, universal gravitational forces, elastic forces and electrical forces is reserved for Physics. A normal force exists between two solid objects when their surfaces are pressed together due to other forces acting on one or both objects (e.g., a solid sitting on or sliding across a table, a ladder leaning against a wall, a ball hitting a bat). A normal force is always a push directed at right angles from the surfaces of the interacting objects. A tension force occurs when a non-slack rope, wire, cord or similar device pulls on another object. In middle school, the concept of a field as a region of space that surrounds objects with the appropriate property (mass for gravitational fields, charge for electric fields, a magnetic object for magnetic fields) was introduced to explain gravitational, magnetic and electrical forces that occur over a distance. In high school, the field concept is further developed. The stronger the field, the greater the force exerted on objects placed in the field. The field of an object is always there even if the object is not interacting with anything else. The gravitational force (weight) of an object is proportional to its mass. Weight, $F_{g},$ can be calculated from the equation $F_{g}=mg,$ where g is the gravitational field strength of an object which is equal to 9.8 N/kg or $9.8~m/s^{2}$ on the surface of Earth.

Forces

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 ...

• Objects at rest 
• Objects moving with constant velocity 
• Accelerating objects

Content elaboration: The focus of the content is to develop a conceptual understanding of the laws of motion to explain and predict changes in motion, not to name or recite a memorized definition. When the vector sum of the forces (net force, Fnet) acting on an object is zero, the object does not accelerate. For an object that is moving, this means the object will remain moving without changing its speed or direction. For an object that is not moving, the object will continue to remain stationary. An object will accelerate (increase or decrease its speed or change its direction of motion) when an unbalanced net force acts on it. The rate at which an object changes its speed or direction (acceleration) is proportional to the vector sum of the forces (net force, $F_{net}$) and inversely proportional to the mass $(a=F_{net}/m)$. These laws will be applied to systems consisting of a single object upon which multiple forces act. Vector addition will be limited to one dimension (positive and negative). While both horizontal and vertical forces can be acting on an object simultaneously, for this level, one of the dimensions must have a net force of zero. A force is an interaction between two objects. Both objects in the interaction experience an equal amount of force, but in opposite directions. Interacting force pairs are often confused with balanced forces. Interacting force pairs can never cancel each other out because they always act on different objects. Naming the force (e.g., gravity, friction) does not identify the two objects involved in the interacting force pair. Objects involved in an interacting force pair can be easily identified by using the format "A acts on B so B acts on A." For example, the truck hits the sign therefore the sign hits the truck with an equal force in the opposite direction. Earth pulls the book down so the book pulls Earth up with an equal force. In Physics, all laws will be applied to systems of many objects.

Dynamics (how forces affect motion)

Vectors, Scalars, and Direction

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