Adding Vectors, Rules

When two vectors are added, the sum is independent of the order of the addition. This is the Commutative

Law of Addition A B B A

Adding Vectors, Rules cont.

When adding three or more vectors, their sum is independent of the way in which the individual vectors are grouped This is called the Associative Property of Addition

A B C A B C

Adding Vectors, Rules final

When adding vectors, all of the vectors must have the same units

All of the vectors must be of the same type of quantity For example, you cannot add a displacement to a

velocity

Negative of a Vector

The negative of a vector is defined as the vector that, when added to the original vector, gives a resultant of zero Represented as

The negative of the vector will have the same magnitude, but point in the opposite direction

A

0 A A

Subtracting Vectors

Special case of vector addition

If , then use Continue with standard

vector addition procedure

A B

A B

Subtracting Vectors, Method 2

Another way to look at subtraction is to find the vector that, added to the second vector gives you the first vector

As shown, the resultant

vector points from the tip of the second to the tip of the first

A B C

Multiplying or Dividing a Vector by a Scalar

The result of the multiplication or division of a vector by a scalar is a vector

The magnitude of the vector is multiplied or divided by the scalar

If the scalar is positive, the direction of the result is the same as of the original vector

If the scalar is negative, the direction of the result is opposite that of the original vector

Component Method of Adding Vectors

Graphical addition is not recommended when High accuracy is required If you have a three-dimensional problem

Component method is an alternative method It uses projections of vectors along coordinate

axes

Components of a Vector, Introduction

A component is a projection of a vector along an axis Any vector can be

completely described by its components

It is useful to use rectangular components These are the projections

of the vector along the x- and y-axes

Vector Component Terminology

are the component vectors of They are vectors and follow all the rules for

vectors

Ax and Ay are scalars, and will be referred to as the components of

x yandA A

A

A

Components of a Vector

Assume you are given a vector

It can be expressed in terms of two other vectors, and

These three vectors form a right triangle

A

xA

yA

x y A A A

Components of a Vector, 2

The y-component is moved to the end of the x-component

This is due to the fact that any vector can be moved parallel to itself without being affected This completes the

triangle

Components of a Vector, 3

The x-component of a vector is the projection along the x-axis

The y-component of a vector is the projection along the y-axis

This assumes the angle θ is measured with respect to the x-axis If not, do not use these equations, use the sides of the

triangle directly

cosxA A

sinyA A

Components of a Vector, 4

The components are the legs of the right triangle whose hypotenuse is the length of A

May still have to find θ with respect to the positive x-axis

2 2 1and tan yx y

x

AA A A

A

Components of a Vector, final

The components can be positive or negative and will have the same units as the original vector

The signs of the components will depend on the angle

Unit Vectors

A unit vector is a dimensionless vector with a magnitude of exactly 1.

Unit vectors are used to specify a direction and have no other physical significance

Unit Vectors, cont.

The symbols

represent unit vectors They form a set of

mutually perpendicular vectors in a right-handed coordinate system

Remember,

kand,j,i

ˆ ˆ ˆ 1 i j k

Viewing a Vector and Its Projections

Rotate the axes for various views

Study the projection of a vector on various planes x, y x, z y, z

Unit Vectors in Vector Notation

Ax is the same as Ax

and Ay is the same as Ay etc.

The complete vector can be expressed as

i

j

ˆ ˆ

x yA AA i j

Adding Vectors Using Unit Vectors

Using Then

and so Rx = Ax + Bx and Ry = Ay + By

ˆ ˆ ˆ ˆ

ˆ ˆ

ˆ ˆ

x y x y

x x y y

x y

A A B B

A B A B

R R

R i j i j

R i j

R i j

2 2 1tan yx y

x

RR R R

R

R A B

Adding Vectors with Unit Vectors

Note the relationships among the components of the resultant and the components of the original vectors

Rx = Ax + Bx

Ry = Ay + By

Three-Dimensional Extension

Using Then

and so Rx= Ax+Bx, Ry= Ay+By, and Rz =Ax+Bz

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ

ˆ ˆ ˆ

x y z x y z

x x y y z z

x y z

A A A B B B

A B A B A B

R R R

R i j k i j k

R i j k

R i j k

2 2 2 1cos , .xx y z

RR R R R etc

R

R A B

Example 3.5 – Taking a Hike

A hiker begins a trip by first walking 25.0 km southeast from her car. She stops and sets up her tent for the night. On the second day, she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger’s tower.

Example 3.5 (A) Determine the components

of the hiker’s displacement for each day.

A

Solution: We conceptualize the problem by drawing a sketch as in the figure above. If we denote the displacement vectors on the first and second days by and respectively, and use the car as the origin of coordinates, we obtain the vectors shown in the figure. Drawing the resultant , we can now categorize this problem as an addition of two vectors.

B A

R

Example 3.5 We will analyze this

problem by using our new knowledge of vector components. Displacement has a magnitude of 25.0 km and is directed 45.0° below the positive x axis.

From Equations 3.8 and 3.9, its components are:

cos( 45.0 ) (25.0 km)(0.707) = 17.7 km

sin( 45.0 ) (25.0 km)( 0.707) 17.7 kmx

y

A A

A A

The negative value of Ay indicates that the hiker walks in the negative y direction on the first day. The signs of Ax and Ay also are evident from the figure above.

A

Example 3.5

The second displacement has a magnitude of 40.0 km and is 60.0° north of east.

Its components are:

cos60.0 (40.0 km)(0.500) = 20.0 km

sin 60.0 (40.0 km)(0.866) 34.6 kmx

y

B B

B B

B

Example 3.5

(B) Determine the components of the hiker’s resultant displacement for the trip. Find an expression for in terms of unit vectors.

Solution: The resultant displacement for the trip has components given by Equation 3.15:

Rx = Ax + Bx = 17.7 km + 20.0 km = 37.7 km

Ry = Ay + By = -17.7 km + 34.6 km = 16.9 km

In unit-vector form, we can write the total displacement as

R

ˆ ˆR = (37.7 + 16.9 ) kmi j

R A B

R

R

Example 3.5

Using Equations 3.16 and 3.17, we find that the resultant vector has a magnitude of 41.3 km and is directed 24.1° north of east.

R

Let us finalize. The units of are km, which is reasonable for a displacement. Looking at the graphical representation in the figure above, we estimate that the final position of the hiker is at about (38 km, 17 km) which is consistent with the components of in our final result. Also, both components of are positive, putting the final position in the first quadrant of the coordinate system, which is also consistent with the figure.

R

R

R

Motion in Two Dimensions

Using + or – signs is not always sufficient to fully describe motion in more than one dimension Vectors can be used to more fully describe motion Will look at vector nature of quantities in more detail

Still interested in displacement, velocity, and acceleration

Will serve as the basis of multiple types of motion in future chapters

Position and Displacement

The position of an object is described by its position vector,

The displacement of the object is defined as the change in its position

r

f ir r r

General Motion Ideas

In two- or three-dimensional kinematics, everything is the same as as in one-dimensional motion except that we must now use full vector notation Positive and negative signs are no longer

sufficient to determine the direction

Average Velocity

The average velocity is the ratio of the displacement to the time interval for the displacement

The direction of the average velocity is the direction of the displacement vector

The average velocity between points is independent of the path taken This is because it is dependent on the displacement, also

independent of the path

avg t

rv

Instantaneous Velocity

The instantaneous velocity is the limit of the average velocity as Δt approaches zero

As the time interval becomes smaller, the direction of the displacement approaches that of the line tangent to the curve

0lim

t

d

t dt

r rv

Instantaneous Velocity, cont

The direction of the instantaneous velocity vector at any point in a particle’s path is along a line tangent to the path at that point and in the direction of motion

The magnitude of the instantaneous velocity vector is the speed The speed is a scalar quantity

Average Acceleration

The average acceleration of a particle as it moves is defined as the change in the instantaneous velocity vector divided by the time interval during which that change occurs.

f i

avgf it t t

v v va

Average Acceleration, cont

As a particle moves, the direction of the change in velocity is found by vector subtraction

The average acceleration is a vector quantity directed along

f iv v v

v

Instantaneous Acceleration

The instantaneous acceleration is the limiting value of the ratio as Δt approaches zero

The instantaneous equals the derivative of the velocity vector with respect to time

0lim

t

d

t dt

v va

tv

Producing An Acceleration

Various changes in a particle’s motion may produce an acceleration The magnitude of the velocity vector may change The direction of the velocity vector may change

Even if the magnitude remains constant Both may change simultaneously

Kinematic Equations for Two-Dimensional Motion

When the two-dimensional motion has a constant acceleration, a series of equations can be developed that describe the motion

These equations will be similar to those of one-dimensional kinematics

Motion in two dimensions can be modeled as two independent motions in each of the two perpendicular directions associated with the x and y axes Any influence in the y direction does not affect the motion

in the x direction

Kinematic Equations, 2

Position vector for a particle moving in the xy plane

The velocity vector can be found from the position vector

Since acceleration is constant, we can also find an expression for the velocity as a function of time:

ˆ ˆx yr i j

ˆ ˆx y

dv v

dt

rv i j

f i tv v a

Kinematic Equations, 3

The position vector can also be expressed as a function of time: This indicates that the position vector is the sum

of three other vectors: The initial position vector The displacement resulting from the initial velocity The displacement resulting from the acceleration

21

2f i it tr r v a

Kinematic Equations, Graphical Representation of Final Velocity

The velocity vector can be represented by its components

is generally not along the direction of either or

fv

iva

Kinematic Equations, Graphical Representation of Final Position

The vector representation of the position vector

is generally not along the same direction as or as

and are generally not in the same direction

fr

iva

fv

fr

Graphical Representation Summary

Various starting positions and initial velocities can be chosen

Note the relationships between changes made in either the position or velocity and the resulting effect on the other