• References– Chapter 2-2.4

VECTORS!

What is a Vector?

Vectors

• A mathematical entity which has a magnitude and direction. Drawn as an arrow.

• In Latin, it means “carrier”– (Diseases, as in rats were a vector for the black

plague)• A few terms:

head

tailLength, magnitude (a scalar)

Notation

• VectorsV

– Sometimes we’ll refer to the components of a Vector like this:

• Vx Vy Vz – Or like this

• Scalars (float’s)k

V

z

y

x

V

V

V

9.3

6.2

3.17

An Example

http://maps.google.com

Another Example

http://w3.shorecrest.org/~lisa_peck/physics/syllabus/mechanics/circularmotion/images_explained.htm

Another

http://www.flickr.com/photos/physicsclassroom/galleries/72157625424161192

Vector components

• X, Y, and Z (in 3D). Mini-offsets along axes.• Example: The vector (5, 3, -1)

1. Start at "current location"2. Using these axes (RHS)…3. Move 5 units (to the right) along the x axis…4. Move 3 units (up) along the y axis…5. Move 1 units (in – WHY?) along the z axis…6. All together it is as if we moved from the origin this distance/direction.

Recap

• Have a magnitude (the length)• Have a direction (the direction)• Composed of mini-offsets along axes.• No position

– Map: Could be from Lucasville to Minford– Coaster: Could be gravity applied to a plane– B.Balls: Could be a car tapping a guard rail.– But…

…They are related.

• One way:– P: Starting position.– V: A vector indicating a direction of movement.– Q: Ending position.

P

V

Q

• Note: The same vector V is the distance and direction from P to Q AND from A to B.

A

V

B

…They are related

• Another way:• The point P = the head of the Vector P when

the tail is at the origin.

• So…that’s why we’ll use our Vector3 class for both Points and Vectors.– We’ll always interpret a vector as one or the other (but

never both at the same time)

Vector P Point P

Vector properties

• We’ll look at a number of Vector Operations (Chapter 5 in the book)

• Ways of looking at each:– Symbolically: With a symbol.– Numerically:

• The steps to “do” this operation• The way you’ll implement it in Python

– Graphically: With a picture• Usually the most complicated / useful• Often many interpretations.

Zero Vector (5.3)

• A property, not an operation• Just a vector with 3 0’s for its components.• Think of it either as:

– A zero-length vector– A point at the origin.

Vector Negation (5.4)

• Symoblically:• Numerically:

– Negate the 3 components– Define the __neg__ method in Python:

• Should return a new Vector3.

• Graphically:

V

V

V

Vector Length (5.5)

• Symbolically: (NOT absolute value)– This is a scalar value.

• Numerically: – How do we do it in 2D?– Pythagorean Theroem!– It’s the same in 3D – just include the z component.– So…– In python, define a length method (not the same

as __len__)

V

222zyx VVVV

Vector Length, graphically

V

V

Vector length, an application

• Idea: If we have a vector indicating the distance between 2 points, we can calculate the length to find the distance between the 2 points.

C

K

V

If the length of V < carRad + kittenRad, the kitten eats the tiny car.

Bounding volume in 3D is a sphere, not a circle.

Vector-scalar multiplication (5.6)

• Symbolically: or or • Numerically:

– Just multiply the components by the scalar to get a new vector.

– In python, define the __mul__ method and the __rmul__ method.

• V * 7 # To python, this is V.__mul__(7).• 7 * V # To python, this is 7.__mul__(V). Int’s don’t know how to

handle a Vector3!• 7 * V # If you’ve define it, this is V.__rmul__(7).

– Just call your __mul__ method.

Vk

Vk

* kV *

Vector-scalar multiplication, cont.

• Graphically: V

5.0*V

V2

Note: Multiplication changes the magnitude, not the direction.

OK, multiplying by a negative reverses the direction – it’s really a combination multiplication & negation

Note2: Point multiplication doesn’t really have a meaning.

)5.0*()5.0(* VV

An Example

Suppose you're travelling along the ground at 10 mph: (0,0,10)

You double your speed: Now your velocity is (0,0,20)

Vector-scalar division

• Symbolically: – Note: isn’t allowed

• Numerically:– Really just (or you can divide the

components)– In python, define the __truediv__ method.

• There is a __rtruediv__ method but DON’T define it!

k

V

V

k

kV1*

Vector normalization (5.7)

• A new symbol:– A Unit or normalized vector is written like

• It’s still a vector.

– Unit-length vectors have the property that

• Normalization is the process of “shrinking” or “growing” a vector so that:– It’s direction is unchanged.– It’s length becomes 1.0

V

0.1ˆ V

Vector normalization, cont.

• Any ideas? What operation “grows” / “shrinks” a vector?– Multiplication (or division).

• Suppose we have a vector V.– Q: What do we need to multiply by so that length

becomes 1.0?– A: The length of the vector.

• So:V

VV

ˆ

Vector Normalization, cont.

V

V

0.1

V

0.1

W

WW

Vector addition & subtraction (5.8)

• Symbolically:

• Numerically:– Addition: Just add the components to get a new

vector (order is not important)– Subtraction: Subtract the components to get a

new vector (order IS important)• The same as • This is a negation followed by an addition

WV

WV

)( WV

Vector Addition, cont.

• Graphically:– Place the second tail on the first head.– The result goes from the first tail to the last head.

V

W

WV

Vector Addition, Example #1

Suppose you jump straight up Let your velocity = (0,4,0)

Represents "jumping, going up at 4 ft/sec"

0,4,0

Example #1, cont.

Consider an elevator that goes up @ 8 ft/sec Let elevator's velocity = (0,8,0)

You ride the elevator up Your speed: (0,8,0)

1TON

Example #1, cont.

Consider: You are riding the elevator and you jump while it's going up

Final speed: Add the vectors (0,8,0) + (0,4,0) = (0,12,0) = 12 ft/sec straight up

Ans: 12 ft/sec straight up Location independent: Doesn't matter where the

elevator is

1TON

Example #1, cont.

Direction matters! Suppose elevator is going DOWN at 8 ft/sec:

(0,-8,0) You jump up: (0,4,0) Final speed: (0,-8,0) + (0,4,0) = (0,-4,0)

.

1TON

Vector Addition, Example #2

• Bird flying east at 4 mph, pushing against SW wind:

• (4,0,0) + (-0.3, -0.7, 0) = (3.7, -0.7, 0)

More on Vector Addition

• It’s commutative:

More on Vector Addition

• You can add more than one Vector:– Just line them up tail-to-head.– The net result goes from the first tail to the last

head.– Still commutative.

V

W

U

UWV

Point Math

Point addition = meaningless Geometrically, you can't add two points

Point + Vector = Move something in a given direction Ex: You are floating in the ocean at (100,4,22) A strong current with direction (40,0,-9) pushes

you Final location: (140,4,13): Add x,y,z components

Vector Subtraction

• Remember, V – W is the same as V + (-W)• So…geometrically:

V

W

W

WV

Point Subtraction

Point – Point: Corresponds to direction to go from one point to another Ex: p = (10,5,0) q=(12,10,0) What is direction from p to q? Take: q-p ← Important! NOT p-q!!! q-p = (12,10,0) – (10,5,0) = (2,5,0) P-q = (10,5,0) – (12,10,0) = (-2,-5,0)

p

q

Point Subtraction Example

T

M

M-T