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Multiplication of vectors and scalars

If we decided to walk three times as far on the first leg of the trip considered in the preceding example, then we would walk ×  27 . 5 m size 12{"3 " times " 27" "." "5 m"} {} , or 82.5 m, in a direction 66 . 0 º size 12{"66" "." 0 { size 12{º} } } {} north of east. This is an example of multiplying a vector by a positive scalar    . Notice that the magnitude changes, but the direction stays the same.

If the scalar is negative, then multiplying a vector by it changes the vector’s magnitude and gives the new vector the opposite direction. For example, if you multiply by –2, the magnitude doubles but the direction changes. We can summarize these rules in the following way: When vector A size 12{A} {} is multiplied by a scalar c size 12{c} {} ,

  • the magnitude of the vector becomes the absolute value of c size 12{c} {} A size 12{A} {} ,
  • if c size 12{A} {} is positive, the direction of the vector does not change,
  • if c size 12{A} {} is negative, the direction is reversed.

In our case, c = 3 size 12{c=3} and A = 27.5 m size 12{"A= 27.5 m"} . Vectors are multiplied by scalars in many situations. Note that division is the inverse of multiplication. For example, dividing by 2 is the same as multiplying by the value (1/2). The rules for multiplication of vectors by scalars are the same for division; simply treat the divisor as a scalar between 0 and 1.

Resolving a vector into components

In the examples above, we have been adding vectors to determine the resultant vector. In many cases, however, we will need to do the opposite. We will need to take a single vector and find what other vectors added together produce it. In most cases, this involves determining the perpendicular components of a single vector, for example the x - and y -components, or the north-south and east-west components.

For example, we may know that the total displacement of a person walking in a city is 10.3 blocks in a direction 29 .0º size 12{"29" "." 0º} } {} north of east and want to find out how many blocks east and north had to be walked. This method is called finding the components (or parts) of the displacement in the east and north directions, and it is the inverse of the process followed to find the total displacement. It is one example of finding the components of a vector. There are many applications in physics where this is a useful thing to do. We will see this soon in Projectile Motion , and much more when we cover forces in Dynamics: Newton’s Laws of Motion . Most of these involve finding components along perpendicular axes (such as north and east), so that right triangles are involved. The analytical techniques presented in Vector Addition and Subtraction: Analytical Methods are ideal for finding vector components.

Phet explorations: maze game

Learn about position, velocity, and acceleration in the "Arena of Pain". Use the green arrow to move the ball. Add more walls to the arena to make the game more difficult. Try to make a goal as fast as you can.

Maze Game

Summary

  • The graphical method of adding vectors A size 12{A} {} and B size 12{B} {} involves drawing vectors on a graph and adding them using the head-to-tail method. The resultant vector R size 12{A} {} is defined such that A + B = R . The magnitude and direction of R size 12{A} {} are then determined with a ruler and protractor, respectively.
  • The graphical method of subtracting vector B from A involves adding the opposite of vector B , which is defined as B size 12{ - B} {} . In this case, A B = A + ( –B ) = R . Then, the head-to-tail method of addition is followed in the usual way to obtain the resultant vector R .
  • Addition of vectors is commutative    such that A + B = B + A size 12{"A + B = B + A"} {} .
  • The head-to-tail method    of adding vectors involves drawing the first vector on a graph and then placing the tail of each subsequent vector at the head of the previous vector. The resultant vector is then drawn from the tail of the first vector to the head of the final vector.
  • If a vector A size 12{A} {} is multiplied by a scalar quantity c size 12{A} {} , the magnitude of the product is given by cA size 12{ ital "cA"} {} . If c size 12{c} {} is positive, the direction of the product points in the same direction as A size 12{A} {} ; if c size 12{c} {} is negative, the direction of the product points in the opposite direction as A size 12{A} {} .

Questions & Answers

basically potentiometer is series circuit or parallel circuit?
muhammad Reply
What is half-life
Godwin Reply
the life in which half of the radioactive element decay
Arif
what is fluid
Anthony Reply
anything that flows is Liquid.
prakash
a substance that has no specific shape
Saleemulhaq
How submarines floats one water the same time sink in water
Courage Reply
A submarine has the ability to float and sink. The ability to control buoyancy comes from the submarine'strim or ballast tanks which can be filled with either water or air, depending on whether the submarine needs to floator sink. When the submarine floats it means its trim tanks are filled with air
Arif
define electromagnetic radiation
Anthony
what is work
Ojo Reply
Force times distance
Karanja
product of force and distance...
Arif
Is physics a natural science?
Adebisi Reply
what is the difference between a jet engine and a rocket engine.
Samuel Reply
explain the relationship between momentum and force
Joseph Reply
A moment is equivalent multiplied by the length passing through the point of reaction and that is perpendicular to the force
Karanja
How to find Squirrel frontal area from it's surface area?
Pooja Reply
how do we arrange the electronic configuration of elements
Muhammed Reply
hi guys i am an elementary student
benedict Reply
hi
Dancan
hello
adolphus
are you an elementary student too?
benedict
no bro
adolphus
yes
Che
hi
Miranwa
yes
Miranwa
welcome
Miranwa
what is the four equation of motion
Miranwa
what is strain?
SAMUEL
Change in dimension per unit dimension is called strain. Ex - Change in length per unit length l/L.
ABHIJIT
strain is the ratio of extension to length..=e/l...it has no unit because both are in meters and they cancel each other
adeleke
How is it possible for one to drink a cold drink from a straw?
Karanja Reply
most possible as it is for you to drink your wine from your straw
Selina
state the law of conservation of energy
Sushma Reply
energy can neither be destroy or created,but can be change from one form to another
dare
yeah
Toheeb
it can neither be created nor destroyed
Toheeb
its so sample question dude
Muhsin
what is the difference between a principle and a law?
Mary Reply
where are from you wendy .?
ghulam
philippines
Mary
why?
Mary
you are beautiful
ghulam
are you physics student
ghulam
laws are ment to be broken
Ge
hehe ghulam where r u from?
Muhsin
yes
dare
principle are meant to be followed
dare
south Africa
dare
here Nigeria
Toheeb
principle is a rule or law of nature, or the basic idea on how the laws of nature are applied.
Ayoka
Rules are meant to be broken while principals to be followed
Karanja
principle is a rule or law of nature, or the basic idea on how the laws of nature are applied.
tathir
what is momentum?
prakash Reply
is the mass times velocity of an object
True
it is the product of mass and velocity of an object.
The momentum possessed by a body is generally defined as the product of its mass and velocity m×v
Usman
momentum is the product of the mass of a body of its velocity
Ugbesia

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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