vectors and scalars a scalar quantity can be described by a single number, with some meaningful unit...

Vectors and scalars

• A scalar quantity can be described by a single number, with some meaningful unit

4 oranges

20 miles

5 miles/hour

10 Joules of energy

9 Volts

Vectors and scalars

• A scalar quantity can be described by a single number with some meaningful unit

• A vector quantity has a magnitude and a direction in space, as well as some meaningful unit.

5 miles/hour North

18 Newtons in the “x direction”

50 Volts/meter down

© 2014 John Wiley & Sons, Inc. All rights reserved.

3-1 Vectors and Their Components

The simplest example is a displacement vector If a particle changes position from A to B, we represent

this by a vector arrow pointing from A to B

Figure 3-1

In (a) we see that all three arrows have the same magnitude and direction: they are identical displacement vectors.

In (b) we see that all three paths correspond to the same displacement vector. The vector tells us nothing about the actual path that was taken between A and B.

Vectors and scalars

• A scalar quantity can be described by a single number with some meaningful unit

• A vector quantity has a magnitude and a direction in space, as well as some meaningful unit.

• To establish the direction, you MUST first have a coordinate system!

Standard x-y Cartesian coordinates common

Compass directions (N-E-S-W)

Drawing vectors

• Draw a vector as a line with an arrowhead at its tip.

• The length of the line shows the vector’s magnitude.

• The direction of the line shows the vector’s direction relative to a coordinate system (that should be indicated!)

x

y

z

5 m/sec at 30 degrees from the x axis towards y in the xy plane

Drawing vectors

• Vectors can be identical in magnitude, direction, and units, but start from different places…

Drawing vectors

• Negative vectors refer to direction relative to some standard coordinate already established – not to magnitude.

Adding two vectors graphically

• Two vectors may be added graphically using either the head-to-tail method or the parallelogram method.

Adding two vectors graphically



The vector sum, or resultanto Is the result of performing vector additiono Represents the net displacement of two or more

displacement vectors

o Can be added graphically as shown:

Figure 3-2

Eq. (3-1)



Vector addition is commutativeo We can add vectors in any order

Eq. (3-2)

Figure (3-3)



Vector addition is associativeo We can group vector addition however we like

Eq. (3-3)

Figure (3-4)



A negative sign reverses vector direction

We use this to define vector subtraction

Eq. (3-4)

Figure (3-5)

Figure (3-6)



These rules hold for all vectors, whether they represent displacement, velocity, etc.

Only vectors of the same kind can be addedo (distance) + (distance) makes senseo (distance) + (velocity) does not



These rules hold for all vectors, whether they represent displacement, velocity, etc.

Only vectors of the same kind can be addedo (distance) + (distance) makes senseo (distance) + (velocity) does not

Answer:

(a) 3 m + 4 m = 7 m (b) 4 m - 3 m = 1 m

Adding more than two vectors graphically

• To add several vectors, use the head-to-tail method.

• The vectors can be added in any order.

Adding more than two vectors graphically—Figure 1.13

• To add several vectors, use the head-to-tail method.

• The vectors can be added in any order.

Subtracting vectors

• Reverse direction, and add normally head-to-tail…

Subtracting vectors

Multiplying a vector by a scalar

• If c is a scalar, the product cA has magnitude |c|A.

Addition of two vectors at right angles

• First add vectors graphically.

• Use trigonometry to find magnitude & direction of sum.

Addition of two vectors at right angles

• Displacement (D) = √(1.002 + 2.002) = 2.24 km

• Direction f = tan-1(2.00/1.00) = 63.4º East of North

Note how the final answer has THREE things!

• Answer: 2.24 km at 63.4 degrees East of North

• Magnitude (with correct sig. figs!)


• Answer: 2.24 km at 63.4 degrees East of North

• Magnitude (with correct sig. figs!)

• Units


• Answer: 2.24 km at 63.4 degrees East of North• Magnitude (with correct sig. figs!)

• Units

• Direction

Components of a vector

• Represent any vector by an x-component Ax and a y-component Ay.

• Use trigonometry to find the components of a vector: Ax = Acos θ and Ay = Asin θ, where θ is measured from the +x-axis toward the +y-axis.

Positive and negative components

• The components of a vector can be positive or negative numbers.

Finding components

• We can calculate the components of a vector from its magnitude and direction.



Components in two dimensions can be found by:

Where θ is the angle the vector makes with the positive x axis, and a is the vector length

The length and angle can also be found if the components are known



In the three dimensional case we need more components to specify a vectoro (a,θ,φ) or (a

x,a

y,a

z)



In the three dimensional case we need more components to specify a vectoro (a,θ,φ) or (a

x,a

y,a

z)

Answer: choices (c), (d), and (f) show the components properly arranged to form the vector

Calculations using components

• We can use the components of a vector to find its magnitude and direction:

• We can use the components of a set of vectors to find the components of their sum:

2 2 and tan yx y

x

AA A A

A

, x x x x y y y yR A B C R A B C

Adding vectors using their components

Unit vectors

• A unit vector has a magnitude of 1 with no units.

• The unit vector î points in the +x-direction, points in the +y-direction, and points in the +z-direction.

• Any vector can be expressed in terms of its components as A =Axî+ Ay + Az .

j

jk

k

j

jk

k


3-2 Unit Vectors, Adding Vectors by Components

A unit vectoro Has magnitude 1o Has a particular directiono Lacks both dimension and unito Is labeled with a hat: ^

We use a right-handed coordinate systemo Remains right-handed when rotated



The quantities axi and a

yj are vector components

The quantities ax and a

y alone are scalar

components

Vectors can be added using components

Eq. (3-9) →



To subtract two vectors, we subtract components

Unit Vectors, Adding Vectors by Components

Eq. (3-13)



To subtract two vectors, we subtract components

Unit Vectors, Adding Vectors by Components

Answer: (a) positive, positive (b) positive, negative

(c) positive, positive

Eq. (3-13)


3-3 Multiplying Vectors

Multiplying two vectors: the scalar producto Also called the dot producto Results in a scalar, where a and b are magnitudes and φ is

the angle between the directions of the two vectors:

The commutative law applies, and we can do the dot product in component form

Eq. (3-20)

Eq. (3-22)

Eq. (3-23)



A dot product is: the product of the magnitude of one vector times the scalar component of the other vector in the direction of the first vector

Eq. (3-21)

Figure (3-18) Either projection of one

vector onto the other can be used

To multiply a vector by the projection, multiply the magnitudes



Answer: (a) 90 degrees (b) 0 degrees (c) 180 degrees

The scalar product

The scalar product of two vectors (the “dot product”) is

A · B = ABcosf

The scalar product

The scalar product of two vectors (the “dot product”) is

A · B = ABcosf

Useful for

• Work (energy) required or released as force is applied over a distance (4A)

• Flux of Electric and Magnetic fields moving through surfaces and volumes in space (4B)

Calculating a scalar product

By components, A · B = AxBx + AyBy + AzBz

Example: A = 4.00 m @ 53.0°, B = 5.00 m @ 130°

Calculating a scalar product

By components, A · B = AxBx + AyBy + AzBz

Example: A = 4.00 m @ 53.0°, B = 5.00 m @ 130°

Ax = 4.00 cos 53 = 2.407

Ay = 4.00 sin 53 = 3.195

Bx = 5.00 cos 130 = -3.214

By = 5.00 sin 130 = 3.830

AxBx + AyBy = 4.50 meters

A · B = ABcos = (4.00)(5.00) f cos(130-53) = 4.50 meters2

The vector product

• The vector product (“cross product”) of two vectors has magnitude

and the right-hand rule gives its direction.

| | sin

ABA B


3-3 Cross Products

o The cross product of two vectors with magnitudes a & b, separated by angle φ, produces a vector with magnitude:

And a direction perpendicular to both original vectors

Direction is determined by the right-hand rule Place vectors tail-to-tail, sweep fingers from the first to

the second, and thumb points in the direction of the resultant vector

Eq. (3-24)



The upper shows vector a cross vector b, the lower shows vector b cross vector a

Figure (3-19)

The vector product

• The vector product (“cross product”) A x B of two vectors is a vector

• Magnitude = AB sin f• Direction = orthogonal (perpendicular) to A and B,

using the “Right Hand Rule”

A

B

A x B

x

y

z



The cross product is not commutative

To evaluate, we distribute over components:

Therefore, by expanding (3-26):

Eq. (3-25)

Eq. (3-26)

Eq. (3-27)



Answer: (a) 0 degrees (b) 90 degrees

The vector cross product

The cross product of two vectors is

A x B (with magnitude ABsinf)

Useful for

• Torque from a force applied at a distance away from an axle or axis of rotation (4A)

• Calculating dipole moments and forces from Magnetic Fields on moving charges (4B)



Angles may be measured in degrees or radians Recall that a full circle is 360˚, or 2π rad

Know the three basic trigonometric functions

Figure (3-11)



Vectors are independent of the coordinate system used to measure them

We can rotate the coordinate system, without rotating the vector, and the vector remains the same

All such coordinate systems are equally valid

Figure (3-15)

Eq. (3-15)

Eq. (3-14)


Scalars and Vectors Scalars have magnitude only

Vectors have magnitude and direction

Both have units!

Adding Geometrically Obeys commutative and

associative laws

Unit Vector Notation We can write vectors in terms

of unit vectors

Vector Components Given by

Related back by

Eq. (3-2)

Eq. (3-5)

Eq. (3-7)

3 Summary

Eq. (3-3)

Eq. (3-6)


Adding by Components Add component-by-component

Scalar Times a Vector Product is a new vector

Magnitude is multiplied by scalar

Direction is same or opposite

Cross Product Produces a new vector in

perpendicular direction

Direction determined by right-hand rule

Scalar Product Dot product

Eqs. (3-10) - (3-12)

Eq. (3-22)

3 Summary

Eq. (3-20)

Eq. (3-24)

vectors and scalars a scalar quantity can be described by a single number, with some meaningful unit...

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