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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

the meaning of phrase in physics
Chovwe Reply
is the meaning of phrase in physics
Chovwe
write an expression for a plane progressive wave moving from left to right along x axis and having amplitude 0.02m, frequency of 650Hz and speed if 680ms-¹
Gabriel Reply
how does a model differ from a theory
Friday Reply
what is vector quantity
Ridwan Reply
Vector quality have both direction and magnitude, such as Force, displacement, acceleration and etc.
Besmellah
Is the force attractive or repulsive between the hot and neutral lines hung from power poles? Why?
Jack Reply
what's electromagnetic induction
Chinaza Reply
electromagnetic induction is a process in which conductor is put in a particular position and magnetic field keeps varying.
Lukman
wow great
Salaudeen
what is mutual induction?
je
mutual induction can be define as the current flowing in one coil that induces a voltage in an adjacent coil.
Johnson
how to undergo polarization
Ajayi Reply
show that a particle moving under the influence of an attractive force mu/y³ towards the axis x. show that if it be projected from the point (0,k) with the component velocities U and V parallel to the axis of x and y, it will not strike the axis of x unless u>v²k² and distance uk²/√u-vk as origin
Gabriel Reply
show that a particle moving under the influence of an attractive force mu/y^3 towards the axis x. show that if it be projected from the point (0,k) with the component velocities U and V parallel to the axis of x and y, it will not strike the axis of x unless u>v^2k^2 and distance uk^2/√u-k as origin
Gabriel Reply
No idea.... Are you even sure this question exist?
Mavis
I can't even understand the question
Ademiye
yes it was an assignment question "^"represent raise to power pls
Gabriel
mu/y³ u>v²k² uk²/√u-vk please help me out
Gabriel
An engineer builds two simple pendula. Both are suspended from small wires secured to the ceiling of a room. Each pendulum hovers 2 cm above the floor. Pendulum 1 has a bob with a mass of 10kg . Pendulum 2 has a bob with a mass of 100 kg . Describe how the motion of the pendula will differ if the bobs are both displaced by 12º .
Imtiaz Reply
no ideas
Augstine
if u at an angle of 12 degrees their period will be same so as their velocity, that means they both move simultaneously since both both hovers at same length meaning they have the same length
Ademiye
Modern cars are made of materials that make them collapsible upon collision. Explain using physics concept (Force and impulse), how these car designs help with the safety of passengers.
Isaac Reply
calculate the force due to surface tension required to support a column liquid in a capillary tube 5mm. If the capillary tube is dipped into a beaker of water
Mildred Reply
find the time required for a train Half a Kilometre long to cross a bridge almost kilometre long racing at 100km/h
Ademiye
method of polarization
Ajayi
What is atomic number?
Makperr Reply
The number of protons in the nucleus of an atom
Deborah
type of thermodynamics
Yinka Reply
oxygen gas contained in a ccylinder of volume has a temp of 300k and pressure 2.5×10Nm
Taheer Reply

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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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