Vector has a magnitude and a direction. Scalar has a magnitude

Vector

has a magnitude and a direction

Scalar

has a magnitude

Vector

has a magnitude and a direction

Scalar

has a magnitude

a brick on a table

Vector

has a magnitude and a direction

Scalar

has a magnitude

brick moves (2 meters)

Vector

has a magnitude and a direction

Scalar

has a magnitude

brick moves

Brick moved 2 meters.

(2 meters)

Vector

has a magnitude and a direction

Scalar

has a magnitude

brick moves

Brick moved 2 meters.

Brick moved

2 meters to the right.

(2 meters)

Vector

has a magnitude and a direction

Scalar

has a magnitude

brick moves

Brick moved 2 meters.

Brick moved

2 meters to the right.

(2 meters)

direction specified

Vector

has a magnitude and a direction

Scalar

has a magnitude

brick moves

Brick moved 2 meters.

Brick moved

2 meters to the right.

(2 meters)

s = “2 m”

s = “2 m to the right”

⃑

“distance”

“displacement”

direction specified

Vector

has a magnitude and a direction

Scalar

has a magnitude

brick moves

Brick moved 2 meters.

Brick moved

2 meters to the right.

(2 meters)

s = “2 m”

s = “2 m to the right”

⃑

“distance”

“displacement”

a vector a scalar

direction specified

Vector Scalar

Some quantities in physics…

displacement s or s

⃑

different notations

Vector Scalar

Some quantities in physics…

displacement s or s velocity v or v

⃑ ⃑

distance s speed, velocity v

Vector Scalar

Some quantities in physics…

displacement s or s velocity v or v acceleration a or a

⃑ ⃑

⃑

distance s speed, velocity v

acceleration a

Vector Scalar

Some quantities in physics…

displacement s or s velocity v or v acceleration a or a

force F or F

⃑ ⃑

⃑ ⃑

distance s speed, velocity v

acceleration a force F

Vector Scalar

Some quantities in physics…

displacement s or s velocity v or v acceleration a or a

force F or F

⃑ ⃑

⃑ ⃑

distance s speed, velocity v

acceleration a force F

time t mass m

Working in 3D

brick moves (2 meters)

The moving brick was a 1D situation…

We will work primarily in 3D in this class.

Working in 3D

brick moves (2 meters)

The moving brick was a 1D situation…

We will work primarily in 3D in this class.

Set up a right-handed coordinate system:

x y

z

x y

z

right-handed system left-handed system

Working in 3D

brick moves (2 meters)

The moving brick was a 1D situation…

We will work primarily in 3D in this class.

Set up a right-handed coordinate system:

x y

z

x y

z

right-handed system left-handed system

The convention is to use a right- handed system.

Displacement in 3D

-3 m in the x direction, and -2 m in the y direction, and -6 m in the z direction

Something moves…

Displacement in 3D

-3 m in the x direction, and -2 m in the y direction, and

-6 m in the z direction

= s

Something moves…

Displacement in 3D

-3 m in the x direction, and -2 m in the y direction, and

-6 m in the z direction

= s

Something moves…

Much better notation…

s = (3 m, 2 m, -6 m)

Displacement in 3D

-3 m in the x direction, and -2 m in the y direction, and

-6 m in the z direction

= s

Something moves…

Much better notation…

s = (3 m, 2 m, -6 m)

More generally…

s ^{= (} s

, s

, s

⁾

“components” of the vector s

x y

z

s ^{= (} s

, s

, s

⁾

Visualizing the components

s

x y

z

s ^{= (} s

, s

, s

⁾

Visualizing the components

s s

s

s

(cf. actual physical model of this set up in class)

x y

z

s ^{= (} s

, s

, s

⁾

Visualizing the components

s s

s

s

(cf. actual physical model of this set up in class)

Magnitude of the vector?

Pythagorean theorem gives it to us…

s = |s| = _⎷ s

⁺ s

⁺ s

A scalar times a vector…

s ^{= (} s

, s

, s

)

Multiplying a vector by a scalar just scales the length of the vector.

as ^{= (} as

, as

, as

⁾

A scalar times a vector…

s ^{= (} s

, s

, s

)

Multiplying a vector by a scalar just scales the length of the vector.

x y

z

as ^{= (} as

, as

, as

⁾ s

2s

Adding vectors

s ^{= (} s

, s

, s

)

Just add components.

 obvious for something like displacement

t ^{= (} t

, t

, t

⁾

s + t = ( s

⁺ t

, s

⁺ t

, s

^+t

⁾

Adding vectors

s ^{= (} s

, s

, s

)

Just add components.

 obvious for something like displacement

t ^{= (} t

, t

, t

⁾

s + t = ( s

⁺ t

, s

⁺ t

, s

^+t

⁾

+ =

Graphically

Unit vectors

Using two preceding rules (for as and s₁+s₂), we can make a new useful notation using unit vectors.

Define three unit vectors:

x

= (1, 0, 0)

y

= (0, 1, 0) z

= (0, 0, 1)

“hat” says that magnitude = 1

Unit vectors

Using two preceding rules (for as and s₁+s₂), we can make a new useful notation using unit vectors.

Define three unit vectors:

x

= (1, 0, 0)

y

= (0, 1, 0) z

= (0, 0, 1)

Can assemble any vector by multiplying and adding these

three vectors together

“hat” says that magnitude = 1

Unit vectors

Using two preceding rules (for as and s₁+s₂), we can make a new useful notation using unit vectors.

Define three unit vectors:

x

= (1, 0, 0)

y

= (0, 1, 0) z

= (0, 0, 1)

Can assemble any vector by multiplying and adding these

three vectors together

s ^{= (} s

, s

, s

⁾ s ⁼ s

^x

⁺ s

^y

⁺ s

^z

equivalent

“hat” says that magnitude = 1

Representation is arbitrary

Or: coordinate system is arbitrary Three identical vectors…

r

r

r

Representation is arbitrary

Or: coordinate system is arbitrary Three identical vectors…

r

r

r

r

⁼ r

⁼ r

Representation is arbitrary

Or: coordinate system is arbitrary Three identical vectors…

r

r

r

r

⁼ r

⁼ r

No particular representation (coordinate system) has been chosen for any of these yet.

Representation is arbitrary

Or: coordinate system is arbitrary Three identical vectors…

r

r

r

r

⁼ r

⁼ r

x y

Choosing a particular representation is often convenient.

s ^{= (} s

, s

, s

⁾

Choosing a particular representation is often convenient.

Also, thinking of the tail of the vector as being at the origin lets you think of vectors as points in 3D space.

s ^{= (} s

, s

, s

⁾

x y

z

s

Choosing a particular representation is often convenient.

Also, thinking of the tail of the vector as being at the origin lets you think of vectors as points in 3D space.

But, take care! Vectors are fundamentally just lengths and directions and aren’t tied to a representation.

s ^{= (} s

, s

, s

⁾

x y

z

s

s

Derivative of a vector

s ⁼ s

^x

⁺ s

^y

⁺ s

z

ds ⁼ ds

⁺ ds

⁺ ds

dt ⁼ dt ^x dt ^y dt ^z

⌵ ⌵ ⌵

+ +

Derivative of a vector

s ⁼ s

^x

⁺ s

^y

⁺ s

z

ds ⁼ ds

⁺ ds

⁺ ds

dt ⁼ dt ^x dt ^y dt ^z

⌵ ⌵ ⌵

If the position of an object is:

x(t)

Then the velocity is:

v(t) ⁼ dx dt

(example on blackboard)

+ +

Multiplying vectors (sort of)

Given these objects called vectors, we can define various useful operations with them.

Multiplying vectors (sort of)

Given these objects called vectors, we can define various useful operations with them.

Two useful operations are multiplication-like in appearance.

dot product and cross product

Multiplying vectors (sort of)

Given these objects called vectors, we can define various useful operations with them.

Two useful operations are multiplication-like in appearance.

dot product and cross product

For each: First, the definition. Then, some intuition.

Dot product

s = s

x

+ s

y

+ s

z

t ^{= t}

^x

⁺ t

^y

⁺ t

z

s⋅t = s

t

+ s

t

+ s

t

Dot product

s = s

x

+ s

y

+ s

z

t ^{= t}

^x

⁺ t

^y

⁺ t

z

s⋅t = s

t

+ s

t

+ s

t

s⋅t = |s| |t| ^{cos 𝜃}

Dot product

s = s

x

+ s

y

+ s

z

t ^{= t}

^x

⁺ t

^y

⁺ t

z

s⋅t = s

t

+ s

t

+ s

t

s⋅t = |s| |t| ^{cos 𝜃}

s⋅s ⁼ |s|

Clearly: ^and

s⋅t ⁼ t⋅s

Dot product – Why?

Gives the product of the magnitudes, with the modification that it only counts the components that are parallel.

a

b

Dot product – Why?

Gives the product of the magnitudes, with the modification that it only counts the components that are parallel.

𝜃

a

b

Dot product – Why?

Gives the product of the magnitudes, with the modification that it only counts the components that are parallel.

a

𝜃

b

b a cos 𝜃

a⋅b = a b cos 𝜃

Dot product – Why?

Can look at it in reverse…

a

𝜃

b

b cos 𝜃 a

a⋅b = a b cos 𝜃

Dot product – Bonus

Can get angle between two vectors knowing only the components without any trouble…

Consider:

s

= (1, 3, -4)

s

= (2, -6, 1)

What’s the angle between these vectors?

Dot product – Bonus

Can get angle between two vectors knowing only the components without any trouble…

Consider:

s

= (1, 3, -4)

s

= (2, -6, 1)

What’s the angle between these vectors?

Equate the two expressions for dot product and solve for cos𝜃

(1)(2) + (3)(-6) + (-4)(1) = [ (1)

⎷

⎷

Cross product

s ⁼ s

^x

⁺ s

^y

⁺ s

^z

t ^{= t}

^x

⁺ t

^y

⁺ t

^z

st ⁼

A cross product yields another vector, perpendicular to the original two.

( s

t

- s

t

^{) x}

⁺ ( s

t

- s

t

^{) y}

⁺ ( s

t

- s

t

^{) z}

Cross product

s ⁼ s

^x

⁺ s

^y

⁺ s

^z

t ^{= t}

^x

⁺ t

^y

⁺ t

^z

st ⁼

A cross product yields another vector, perpendicular to the original two.

st ⁼ |s| |t| ^{sin 𝜃}

^u

Alternative form. Here, the

vector u points perpendicular to the plane of s and t, according to a right-hand rule…

( s

t

- s

t

^{) x}

⁺ ( s

t

- s

t

^{) y}

⁺ ( s

t

- s

t

^{) z}

⌵ ^⌵

Cross product

s ⁼ s

^x

⁺ s

^y

⁺ s

^z

t ^{= t}

^x

⁺ t

^y

⁺ t

^z

st ⁼

A cross product yields another vector, perpendicular to the original two.

st ⁼ |s| |t| ^{sin 𝜃}

^u

Alternative form. Here, the

vector u points perpendicular to the plane of s and t, according to a right-hand rule…

st = -ts

Not commutative!

( s

t

- s

t

^{) x}

⁺ ( s

t

- s

t

^{) y}

⁺ ( s

t

- s

t

^{) z}

⌵ ^⌵

a

b

Cross product – Why?

𝜃

Gives the product of the magnitudes with the modification that it neglects any parallel components.

a sin 𝜃

b

vector pointing out of page vector pointing into page

ab

⊗

⊙

⊗

|ab| = a b sin 𝜃

Cross product – Bonus

If you are familiar with matrices and determinants…

x

ab =

y z

a

a

a

b

b

b

Cross product – Bonus

If you are familiar with matrices and determinants…

x

ab =

y z

a

a

a

b

b

b

For the curious:

The result of a cross product is subtly different from a regular

vector and is sometimes called a “pseudovector” or “axial vector”.

(further discussion in class)