Vectors in Two

Dimensions

Read Chapter 1.6-1.9

Scalars and VectorsAll measurements are considered to be

quantities. In physics, there are 2 types of quantities – SCALARS AND VECTORS.

Scalar quantities have only magnitude.

Vector quantities have magnitude and direction.

time mass

temperature

displacement

velocity

acceleration

Gravitational Field

Magnetic Field

Force

Vectors are used to describe motion and solve problems concerning motion.

For this reason, it is critical that you have an understanding of how to

represent vectors add vectorssubtract vectorsmanipulate vector

quantities.in 2 and 3 dimensions.

Vectors

tail

tip

Magnitude represented by the length of the vector

8 units5 units2 units

Vectors

Direction represented by the direction of the arrow

x

y

003008001200600 from -x2250

450

from-y

3000

-600

Adding VectorsWe know how to add vectors in 1-dimension.

Example: displacement. If someone walks 4 mi east (Dx1) and then 7 mi west (Dx2) the total displacement (Dx1 + Dx2) is 3 mi west.1. Adding vectors GRAPHICALLY – place them TAIL TO TIP

2. Adding vectors MATHEMATICALLY – In 1- dimension, assign direction + or – and add algebraically

Dx1 + Dx2 = +4 mi + (-7 mi) = -3 mi

east

+west

-

Dx1 = 4 mi

Dx2 = 7 mi

Dx1 + Dx2 = 3 mi

Adding VectorsWhat about if the vectors are in

different directions in 2-D?How do we describe the direction?How do we add/subtract the vectors?

v 1 =

5

v2 = 350o above +x

35o below +x

Adding Vectors Graphically“tail to tip”To add vectors using the

tail to tip method

1. Draw the first vector (7u, 50o N of E) beginning at the origin.

2. Draw the second vector (3u, 35o S of E) with its tail at the tip of the first vector.

3. Draw the Resultant vector (the answer) from the tail of the first vector to the tip of the last.

North

South

EastWest

v1

v2

v = v 1 + v 2

Adding Vectors Grahically

“parallelogram method”To add vectors with the parallelogram method

1.Draw the first vector to scale beginning at the origin.

2.Draw the second vector, to scale, with its tail also at the origin.

3.Starting at the tip of one vector, draw a dotted line parallel to the other vector. Repeat, starting from the tip of the second vector.

4.Draw the Resultant vector (the answer) from the origin to the intersection of the dotted lines.

North

South

EastWest

v1

v2

v = v 1 + v 2

Adding Vectors MathematicallyWhat if the vectors are in different directions?

For example, what if I walk 5 steps north and then 4 steps east. What is my total displacement , Dx, for the trip?

OR what is the vector sum of the Dx1 and Dx2?

Dx2 =4 steps east

Dx1=5 steps north Dx = Dx1+Dx2 = ?

Adding Vectors Mathematically4 steps east

5 steps north Dx = ?

Use Pythagorean Theorem to find the magnitude of Dx

steps 4.64145 222

xx

Use right triangle trig to find the direction of Dx

north ofeast 7.38)8.0(tan

8.05

4tan

0

1

q

RIGHT TRIANGLE TRIGONOMETRY

A

OH

q

A – side adjacent to angle qO – side opposite to angle qH – hypotenuse of triangle

sinq = OH

cosq = AH

tanq = sinqcosq =

OA

Pythagoreon Theorem 222 AOH

SOHCAHTOA

A student walks a distance of 240 m East, then walks 150m south in 30 min. What is the net displacement? What is the average velocity for the trip?

West

tanq = oppadj

150m240m=

tanq = 0.625q= tan-1(1.3) = 32o south of east

MAGNITUDE R2 = A2 +B2

= 2402 + 1502

R = 283 mDIRECTION

North

South

EastA = 240 m

B=150m

qR=283 m

Notice that A and B are perpendicular components of R. They are the amount of R in each direction

A student walks a distance of 240 m East, then walks 150m south in 30 min. What is the net displacement? What is the average velocity for the trip?

West

North

South

East240 m

150mq283 m

The net displacement is 283 m in the direction of 32o S of E.

The average velocity:

v = DxDt

283 m0.5 hr=

= 566 m/hr in the direction 32o S of E.

EXAMPLE: What is the Resultant of adding 2 vectors, A and B, if A = 8 units south and B = 4.5 u west?

North

South

EastWest

4.5u west

8u southq

tanq = oppadj

4.5 u8 u=

tanq = 0.56

q-1 = tan-1(0.56) = 29o 9.2 u

The Resultant is 9.2 units in the direction of 29o south of west

MAGNITUDE R2 = A2 +B2

= 82 + 4.52

R = 9.2 uDIRECTION

R

Notice that A and B are perpendicular components of R. They are the amount of R in each direction

30o

Ex. Vector A has magnitude 8.0 m at an angle of 30 degrees below the x-axis. What are the x- and y-components of A?

A=8Ay

Ax

sin30 = Ay

AAy = 8sin30 = 8(0.5) = - 4m

cos30 = Ax

AAx = 8cos30 = 8(0.866) = +6.9

Perpendicular Components of a Vector Any vector can be resolved into perpendicular components. Use right triangle trig – x and y components always make a right triangle with the vector .

Ax = Acos30

Ay = Asin30

You must assign the correct direction

What are the x- and y- components of the vector A, shown below?

A = 8 m

30o x

y

Ax

Ay Ax = Acosq

= 8cos(30)

= 8(0.866)

= -6.9 m Ay = Asinq

= 8sin(30)

= 8(0.5)

= +4 m

x- and y- Components of a Vector

Ax is the amount of A in the x-direction Ay is the amount

of A in the y-direction

North

South

East

West ?q

A butterfly moves with a speed of 12 m/s. The x-component of its velocity is 8.00 m/s. The angle between the direction of its motion and the x-axis must be what?

v = 12 m/s

vy

vx = 8cosq = vx

v

q = cos-10.667 = 48o

812=

Adding Vectors by ComponentsWhat about adding 2 vectors, A and B, that are NOT

perpendicular or parallel?

y

x

Ry

60o

A

B By

Bx

R

Rx

Any vector can be described as the sum of perpendicular components.

Components in the same direction are added as 1-D vectors to find the components of the resultant vector.

Rx = Ax +

Bx

= 0 + Bx

Ry = Ay +

By

(Ax+Ay)+(Bx+By)=(Rx+Ry)

A + B =

R

Adding x- and y- Components of a Vector

Ry

60o

A

B By

Bx

R

Rx

R2 = Rx2 + Ry

2

x Y

A Ax Ay

B Bx By

R Rx Ry

A + B =

R Determine the perpendicular components of each vector. Make a table to add up x and y components separately:

q

Magnitude of R:

x

y

x

y

R

R

R

R 1tantan Direction of R:

(Ax+Ay)+(Bx+By)=(Rx+Ry)

y

x

ADDING VECTORS GRAPHICALLYYou can add as many vectors as

you wantNorth

South

West

B

A

R

C

D

East

A + B + C + D =

R Graphically, the vectors are added “tail to tip”

and the order doesn’t matter

C + A + D+ B = R

A

B C

D

A + B + C + D =

R To Add Vectors by Components:

1.Place each vector at origin. Find the x- and y- components of each vector in the sum, and list them in a table. Make sure to include the direction of each component by hand.

2.Add all the x-components to find Rx.

3.Add all the y-components to find Ry.

4.Use Pythagorean Theorem and trigonometry to find the magnitude and direction of R.

Mathematically ADDING VECTORS (Ax+Ay)+(Bx+By)+(Cx+Cy)+

(Dx+Dy)=(Rx+Ry) x Y

A -Ax -Ay

B+B

x-By

C+C

x

+Cy

D -Dx+D

y

R

Rx Ry

R2 = Rx2 + Ry

2Magnitude of R:

x

y

R

R1tan Direction of R:

EXAMPLE (Adding Vectors by Components)

Determine the resultant of the following 3 displacements:

A. 24m, 30º north of eastB. 28m, 37º east of northC. 20m, 50º west of south

x (m) y (m)

A 20.8 12.0

B 16.9 22.4

C -15.3 -12.9

Ʃ 22.4 21.5

North

South

West

A B

C

East

30o

37o

50o

Ay

Ax

By

Bx

Cy

Cx

EXAMPLE

x (m) y (m)

A 20.8 12.0

B 16.9 22.4

C -15.3 -12.9

Ʃ R 22.4 21.5

North

South

West

R

East

m315.214.22 22

22

222

yx

yx

RRR

RRR

Ry

Rx

Magnitude Direction

01 4496.0tan4.22

5.21tan

x

y

R

R

q

R = 31 m, 440 N of E (from the x-axis)

DO NOWAn airplane trip involves 3 legs, with 2 stopovers. The first leg is due east for 620 km, the second is southeast (-450) for 440 km, and the 3rd leg is at 530, south of west, for 550 km. What is the plane’s total displacement?

North

South

West A

B C

East

45o53o By Cy

Cx Bx

x (km)

y (km)

A 620 0

B 311 -311

C -331 -439

DR 600 -750

DR

EXAMPLE (cont.)North

South

West

DR

East

q

x (km)

y (km)

A 620 0

B 311 -311

C -331 -439

DR 600 -750

m960750600 22

222

kR

RRR yx

Magnitude Direction

01 51)25.1(tan600

750tan

x

y

R

R

DR = 960 km, -510 from the x-axis (510 S of E)

EXAMPLEAn airplane trip involves 3 legs, with 2 stopovers. The first leg is due east for 620 km, the second is southeast (-450) for 440 km, and the 3rd leg is at 530, south of west, for 550 km. What is the plane’s total displacement?

North

South

West

DR

East

q

960 km at 510 South of East

Subtracting VectorsIn order to subtract a vector, we add the negative of that vector.

The negative of a vector is defined as a vector in the OPPOSITE direction (with each component the negative of the original)v1 v2- v1 -v2+=

v1

-v2

= v1

v2

-

A=8 B=10

60o

A + B = S

A – B = D

AB

S A

B D

D -B

A

Graphical Representation

Tail to tip Tail to tip(A and –B)

Tail to tail(A and B)

-B60o

A=8 B=10

60o

A + B = S

A + (–B) = D

Mathematical Representation

x y

A + 8 0 B + 5 - 8.7

S 13 -8.7

x y

A + 8 0

-B - 5+

8.7

D 3 +8.7

S

Sx

Syq D

Dx

Dyq

-B

60o

Scalar MultiplicationMultiplication of a vector by a positive scalar changes

the magnitude of the vector, but leaves its direction unchanged. The scalar changes the size of the vector. The scalar "scales" the vector.

Multiplication of a vector by a negative scalar changes the magnitude of the vector, and makes the direction opposite.

Example:

3A = 3 x = =

If the scalar is negative, it changes the direction of the vector.

-3A = -3 x = =