New concepts: scalars, vectors, unit

vectors, vector components, vector

equations, scalar productreading assignment read chap 3

Most physical quantities are described by a single number or variableexamples: your age, your weight, today’s temperature, the time, thecolor (frequency of light) of your car etc.

The above are called scalars.

Some physical quantities are better described by 2 or more numbersor variables.

examples: displacement in 2 and 3 dimensions, you need magnitude and direction (two or more numbers) to completely describe it.

These are called vectors, objects that require magnitude and direction

Physical quantities that are vectors: include displacement, velocity, acceleration, force in 2 and 3 dimensions

Vectors, are more convenient and compact mathematical notation.

Example. Suppose we want to describe the displacement from Glenwood Springs to Fort Collins. (note this is TWO dimensionaldisplacement).

We could say, the displacement is 100 miles East and 60 miles North.This is convenient for an automobile.

OR we can represent this displacement bya vector drawn in red, denoted as A.

To fully describe the vector we needto know is length (magnitude) andits direction ( in this case theangle w.r.t. horizontal ).This is useful for airplanes Glenwood Springs

Fort Collins

Denver

A

x axis

y axis

θ

magnitudeor

length

displacement vector

A

B

displacement vector from point A to B can be describedby a magnitude (or length) and direction θ

terminology: vector quantities are boldface characters, aolder texts use, , regular font a or |a| is magnitude of vector ara a or

UNIT VECTORSsuppose we make unit vectors, 1 unit magnitude in the x direction, i ,and 1 unit magnitude in the y direction, j .

X

Y

x unit vectors, i

y unit vectors j

a = i + i + i + i + i + j + j + j = 5 i + 3 j

a

we can write vector a as a sum of vectors, in this case we add5 i vectors and 3 j vectors. vector components of a are ax and ay.ax =5 and ay =3 and we can write a = ax i + ay j.

Self Test Question;

Suppose we have vector A = 5 i and vector B = 5 j

What is the magnitude of vector, C=A + B , or | A + B | ??

Ans;

What is the direction θ of the vector, A + B ?

Ans;

A

B

Self Test Question;

Suppose we have vector A = 5 i and vector B = 5 j

What is the magnitude of vector, C= A + B , or | A + B | ??

Ans; | A + B | = sqrt ( 52 + 52 ) = sqrt(50)

What is the direction θ of the vector, A + B ?

Ans; θ = 45°

A

BC=A + B

Self Test Question;

Suppose we have vector A = 5 i and vector B = 5 j

What is the magnitude of vector, C= A + B , or | A + B | ??

Ans; | A + B | = sqrt ( 52 + 52 ) = sqrt(50)

What is the direction θ of the vector, A + B ?

Ans; θ = 45°

What are the x and y components of C ?

Ans; A

BC=A + B

Self Test Question;

Suppose we have vector A = 5 i and vector B = 5 j

What is the magnitude of vector, C= A + B , or | A + B | ??

Ans; | A + B | = sqrt ( 52 + 52 ) = sqrt(50)

What is the direction θ of the vector, A + B ?

Ans; θ = 45°

What are the x and y components of C ?

Ans; Cx = 5, Cy = 5A

BC=A + B

We can write vectors in terms of magnitude and directionor in terms of the x component and the y component.

What is the relation between these two different sets of variables?

magnitude = = |a| = a

tanθ = ay/ax or atan(ay/ax)=θ

ax = a cosθ

ay = a sinθ

a ax y2 2+

θax

ay

length a

a = b + c + d + e

adding vectors graphically, place the origin of one vector onthe arrow tip of another

a

b

c

d

e

Multiplying vectors by real numbers (scalar), magnitude changesbut direction does not.

=

i + i + i + i + i + i = 6 i = a

Multiplying vector by a negative number, reverses direction, mag. same

b = − a = − 6 i

Subtracting vectors

a + b

ab a

-ba - b

Scalar Product or dot product

In Physics we will need to form a scalar quantity formed from two vectors. Later we will use them in Work and Electric fields

definition: a ⋅ b = a b cosθ , where θ is the angle between vectors.

example, i ⋅ i = 1 × 1 × cos 0° = 1 , i ⋅ j = 1 × 1 × cos90° =0

if a = ax i + ay j and b = bx i + by j.

a ⋅ b = (ax i + ay j ) ⋅ (bx i + by j) =

a ⋅ b = ax bx i ⋅ i + ax by i ⋅ j + ay bx j ⋅ i + ay by j ⋅ j

a ⋅ b = ax bx + ay by

Useful things to do with unit vectors and dot products.

given any vector a, we can obtain the components by using the dotproducts with the unit vectors.

a⋅ i =( ax i + ay j ) ⋅ i = ax i ⋅ i = ax

We can say that the dot product projects the vector component in the direction of the unit vector. That is the dot product of a unit vector anda given vector will yield the vector component of the given vector.

If we multiply a vector by itself then,

a⋅ a = ( ax i + ay j ) ⋅ ( ax i + ay j ) = ax2+ ay

2 = a2

hence the length of a is sqrt. root of a⋅ a

Problem: A plane is to fly due north. The speed of the plane relativeto the air is 200 km/h, and the wind is blowing from the west to east at 90 km/h.(a) in which direction should the plane head?(b) how fast does the plane travel relative to the ground?

Step 1 draw Diagram

Problem: A plane is to fly due north. The speed of the plane relativeto the air is 200 km/h, and the wind is blowing from the west to east at 90 km/h.(a) in which direction should the plane head?(b) how fast does the plane travel relative to the ground?

N

W

vfinal

vplane

vwind

θ

Problem: A plane is to fly due north. The speed of the plane relativeto the air is 200 km/h, and the wind is blowing from the west to east at 90 km/h.(a) in which direction should the plane head?(b) how fast does the plane travel relative to the ground?

N

W

vfinal

vplane

vwind

θSolution step by step(1) final velocity eqn. is ?

Problem: A plane is to fly due north. The speed of the plane relativeto the air is 200 km/h, and the wind is blowing from the west to east at 90 km/h.(a) in which direction should the plane head?(b) how fast does the plane travel relative to the ground?

N

W

vfinal

vplane

vwind

θSolution step by step(1) final velocity eqn. is ? vfinal=vplane+ vwind

(2) what is the angle:

Problem: A plane is to fly due north. The speed of the plane relativeto the air is 200 km/h, and the wind is blowing from the west to east at 90 km/h.(a) in which direction should the plane head?(b) how fast does the plane travel relative to the ground?

N

W

vfinal

vplane

vwind

θSolution step by step(1) final velocity eqn. is ? vfinal=vplane+ vwind

(2) what is the angle: sine of the angle θ between the velocity of the plane and north equals the ratios vwind and vplane

sinθ = 90km/h / 200km/h = 0.45, θ = 26.7

(3) what is the plane velocity?

Problem: A plane is to fly due north. The speed of the plane relativeto the air is 200 km/h, and the wind is blowing from the west to east at 90 km/h.(a) in which direction should the plane head?(b) how fast does the plane travel relative to the ground?

N

W

vfinal

vplane

vwind

θSolution step by step(1) final velocity eqn. is ? vfinal=vplane+ vwind

(2) what is the angle: sine of the angle θ between the velocity of the plane and north equals the ratios vwind and vplane

sinθ = 90km/h / 200km/h = 0.45, θ = 26.7

(3) what is the plane velocity?since vfinal and vwind are perpendicular, we use the Pythagorean Theorem to find the magnitude of vfinal, vplane

2= vwind2 + vfinal

2

vfinal= sqrt (vplane2 - vwind

2 )

Self testA jet plane in straight horizontal flight passes over your head. When it is directly above you, the sound seems to come from a point behind the plane in a direction 30° from the vertical. The speed of the plane is:

A) the same as the speed of sound B) half the speed of sound C) three-fifths the speed of sound D) 0.866 times the speed of sound E) twice the speed of sound

Hint Draw Picture and write down relevant features of the problem

2 minutes

30°

Distance jet travels= vJET t

Distance= vsound t

Solution:0.5 = sin(30°) = vJET t / vsound t = vJET / vsound

Ans) B

Distance sound travels