10.2 day 2 Vector Valued Functions

Greg Kelly, Hanford High School, Richland, WashingtonPhoto by Vickie Kelly, 2006

Everglades National Park, FL

Photo by Vickie Kelly, 2006

Everglades National Park, FL

Photo by Vickie Kelly, 2006

Everglades National Park, FL

Photo by Vickie Kelly, 2006

Everglades National Park, FL

Any vector can be written as a linear

combination of two standard unit vectors.

,a bv

1,0i 0,1j

,a bv

,0 0,a b

1,0 0,1a b

a b i j

The vector v is a linear combination

of the vectors i and j.

The scalar a is the horizontal

component of v and the scalar b is

the vertical component of v.

We can describe the position of a moving particle by a vector, r(t).

tr

If we separate r(t) into horizontal and vertical components,

we can express r(t) as a linear combination of standard unit vectors i and j.

t f t g t r i j f t i

g t j

In three dimensions the component form becomes:

f g ht t t t r i j k

Graph on the TI-89 using the parametric mode.

MODE Graph……. 2 ENTER

Y= ENTERxt1 t cos t

yt1 t sin t

WINDOW

GRAPH

cos sin 0t t t t t t r i j

8

Graph on the TI-89 using the parametric mode.

cos sin 0t t t t t t r i j

MODE Graph……. 2 ENTER

Y= ENTERxt1 t cos t

yt1 t sin t

WINDOW

GRAPH

Most of the rules for the calculus of vectors are the same as we have used, except:

Speed v t

velocity vectorDirection

speed

t

tv

v

“Absolute value” means “distance from the origin” so we must use the Pythagorean theorem.

Example 5: 3cos 3sint t t r i j

a) Find the velocity and acceleration vectors.

3sin 3cosd

t tdt

r

v i j

3cos 3sind

t tdt

v

a i j

b) Find the velocity, acceleration, speed and direction of motion at ./ 4t

Example 5: 3cos 3sint t t r i j

3sin 3cosd

t tdt

r

v i j 3cos 3sind

t tdt

v

a i j

b) Find the velocity, acceleration, speed and direction of motion at ./ 4t

velocity: 3sin 3cos4 4 4

v i j

3 3

2 2 i j

acceleration: 3cos 3sin4 4 4

a i j

3 3

2 2 i j

Example 5: 3cos 3sint t t r i j

3sin 3cosd

t tdt

r

v i j 3cos 3sind

t tdt

v

a i j

b) Find the velocity, acceleration, speed and direction of motion at ./ 4t

3 3

4 2 2

v i j

3 3

4 2 2

a i j

speed:4

v

2 23 3

2 2

9 9

2 2 3

direction:

/ 4

/ 4

v

v3/ 2 3/ 2

3 3

i j

1 1

2 2 i j

Example 6: 3 2 32 3 12t t t t t r i j

2 26 6 3 12d

t t t tdt

r

v i j

a) Write the equation of the tangent where .1t

At :1t 1 5 11 r i j 1 12 9 v i j

position: 5,11 slope:9

12

tangent: 1 1y y m x x

311 5

4y x

3 29

4 4y x

3

4

The horizontal component of the velocity is .26 6t t

Example 6: 3 2 32 3 12t t t t t r i j

2 26 6 3 12d

t t t tdt

r

v i j

b) Find the coordinates of each point on the path where the horizontal component of the velocity is 0.

26 6 0t t 2 0t t

1 0t t 0, 1t

0 0 0 r i j

1 2 3 1 12 r i j

1 1 11 r i j

0,0

1, 11