phy10 vectors

Vectors

Engr. E. D. Dimaunahan

1. The system of units used by scientists and engineers around the world is commonly called?

2. Write the equivalent of the following Greek Prefixes?Micro, mili, kilo, nano

3. Differentiate a scalar and vector quantities.

4. Other name for vector sum.

5. Give examples of scalar and vector quantities.

Vectors and Their Components

Physics deals with quantities that have both size and direction

A vector is a mathematical object with size and direction

A vector quantity is a quantity that can be represented by a vector

Examples: position, velocity, acceleration Vectors have their own rules for manipulation

A scalar is a quantity that does not have a direction Examples: time, temperature, energy, mass Scalars are manipulated with ordinary algebra

VECTORS

resultant vector – the vector sum of two or more concurrent vectors.

equilibrant vector – a single vector that balances two or more concurrent vectors.

equivalent vectors – two vectors that have the same magnitude and direction.

parallel vectors – two vectors that have the same direction.

anti-parallel vectors – two vectors that have opposite directions.

negative of a vector – a vector that has the same magnitude as but opposite the direction of another vector.

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

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 Their Components

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:

72.4 m, 32 degrees east of north57.3 m, 36 degrees south of west17.8 m straight south

Vectors and Their Components

Vector addition is commutativeo We can add vectors in any order

Eq. (3-2)

Figure (3-3)

Vectors and Their Components

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

Figure (3-4)

Vectors and Their Components

A negative sign reverses vector direction

We use this to define vector subtraction

Eq. (3-4)

Vectors and Their Components

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

Vectors and Their Components

Rather than using a graphical method, vectors can be added by componentso A component is the projection of a vector on an axis

The process of finding components is called resolving the vector

The components of a vector can be positive or negative.

They are unchanged if the vector is shifted in any direction (but not rotated).

Vectors and Their Components

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

Therefore, components fully define a vector

Vectors and Their Components

In the three dimensional case we need more components to specify a vector

o (a,θ,φ) or (ax,a

y,a

z)

Vectors and Their Components

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)

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

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

Unit Vectors, Adding Vectors by Components

The quantities axi and a

yj are vector components

The quantities ax and a

y alone are scalar components

o Or just “components” as before

Vectors can be added using components

Eq. (3-9) →

Unit Vectors, Adding Vectors by Components

To subtract two vectors, we subtract components

Unit Vectors, Adding Vectors by Components

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

(c) positive, positive

Unit Vectors, Adding Vectors by Components

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

Multiplying Vectors

Multiplying a vector z by a scalar co Results in a new vectoro Its magnitude is the magnitude of vector z times |c| o Its direction is the same as vector z, or opposite if c is

negativeo To achieve this, we can simply multiply each of the

components of vector z by c

To divide a vector by a scalar we multiply by 1/cExample Multiply vector z by 5

o z = -3 i + 5 jo 5 z = -15 i + 25 j

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

Multiplying Vectors

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)

Either projection of one vector onto the other can be used

To multiply a vector by the projection, multiply the magnitudes

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

Multiplying Vectors

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

Multiplying Vectors

Multiplying two vectors: the vector producto The cross product of two vectors with magnitudes a & b,

separated by angle φ, produces a vector with magnitude:

o 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)

Multiplying Vectors

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

Multiplying Vectors

The cross product is not commutative

To evaluate, we distribute over components:

Therefore, by expanding:

Multiplying Vectors

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

UNIT VECTOR

A unit vector is a vector with a magnitude of exactly 1 and points in a particular direction.

+x

+y

+z

i

j

k

A vector in its “regular” form or “magnitude-angle” form, such as

xabovemA o 30,00.10

can be expressed in “unit vector” form or “rectangular” form.

jmimA ˆ)00.5(ˆ)66.8(

+x

+y

+z

iA xˆ

kA zˆ

jA yˆ

A

VECTOR ADDITION

+x

+y

+z

B

iB xˆ

kB Zˆ

jB yˆ

kmjmimA ˆ)6(ˆ)8(ˆ)10(

kmjmimB ˆ)4(ˆ)12(ˆ)8(

kmjmimA ˆ)6(ˆ)8(ˆ)10(

kmjmimB ˆ)4(ˆ)12(ˆ)8(

kmjmimC ˆ)10(ˆ)20(ˆ)2(

BAC

Find the resultant of the following vectors.

kmjmimA ˆ)6(ˆ)8(ˆ)10(

kmjmimB ˆ)4(ˆ)12(ˆ)8(

VECTOR PRODUCTS: Scalar/Dot Product

kAjAiAA zyxˆˆˆ

kBjBiBB zyxˆˆˆ

Given the following vectors:

The scalar (or dot) product of these vectors is defined as:

cosABBA

Where: A is the magnitude of A

B is the magnitude of B

is the angle between and .A

B

VECTOR PRODUCTS: Scalar/Dot Product

cosABBA

10cos)1)(1(ˆˆ ii

090cos)1)(1(ˆˆ ji

090cos)1)(1(ˆˆ ki

090cos)1)(1(ˆˆ ij

10cos)1)(1(ˆˆ jj

090cos)1)(1(ˆˆ kj

090cos)1)(1(ˆˆ ik

090cos)1)(1(ˆˆ jk

10cos)1)(1(ˆˆ kk

VECTOR PRODUCTS: Scalar/Dot Product

kBjBiBkAjAiABA zyxzyxˆˆˆˆˆˆ

kBjBiBB zyxˆˆˆ

kAjAiAA zyxˆˆˆ

zzyyxx BABABABA

cosABBA

Find the dot product of the following vectors:

kmjmimA ˆ)6(ˆ)8(ˆ)10(

kmjmimB ˆ)4(ˆ)12(ˆ)8(

zzyyxx BABABABA

)]4)(6[()]12)(8[()]8)(10[( mmmmmmBA

222 249680 mmmBA

240mBA

Find the angle between the following vectors:

kmjmimA ˆ)6(ˆ)8(ˆ)10(

kmjmimB ˆ)4(ˆ)12(ˆ)8(

cosABBA

cos22420040

224200

40cos 1

o11.79

VECTOR PRODUCTS: vector/Cross Product

kAjAiAA zyxˆˆˆ

kBjBiBB zyxˆˆˆ

Given the following vectors:

The vector (or cross) product of these vectors is defined as:

sinABBA

Where: A is the magnitude of A

B is the magnitude of B

is the angle between and .A

B

VECTOR PRODUCTS: Vector/Cross Product

sinABBA

00sin)1)(1(ˆˆ ii

kji ˆ90sin)1)(1(ˆˆ

jki ˆ90sin)1)(1(ˆˆ

kij ˆ90sin)1)(1(ˆˆ

00sin)1)(1(ˆˆ jj

ikj ˆ90sin)1)(1(ˆˆ

jik ˆ90sin)1)(1(ˆˆ

ijk ˆ90sin)1)(1(ˆˆ

00sin)1)(1(ˆˆ kk

kBjBiBkAjAiABA zyxzyxˆˆˆˆˆˆ

kBjBiBB zyxˆˆˆ

kAjAiAA zyxˆˆˆ

VECTOR PRODUCTS: Vector/Cross Product

sinABBA

kBABAjBABAiBABABA xyyxzxxzyzzyˆˆˆ

Find the cross product of the following vectors:

kmjmimA ˆ)6(ˆ)8(ˆ)10(

kmjmimB ˆ)4(ˆ)12(ˆ)8(

kBABAjBABAiBABABA xyyxzxxzyzzyˆˆˆ

kjiBA ˆ)64(120ˆ4048ˆ7232

kmjmimBA ˆ184ˆ88ˆ40 222

PROBLEM:

PROBLEM:

In the figure at right, a cube is placed so that one corner (point O) is at the origin and three edges are along the +x +y, and +z axes of a coordinate system. Use unit vectors to find the  (a) angle between the edge along the z-axis (line OQ) and the diagonal of the cube (line OP). (b) angle between the diagonal of a face (line OR) and line OP.