Vectorsscalar quantities

• magnitude

• mass• temperature• electric potential

vectors• magnitude &

direction• bold face in text

• force• heat flux• electric field• magnetic field

AAuA

AAu

2z

2y

2x AAA A

zyx uuuA 1243

131243 222 A

131243 zyx

Auuu

u

AAuA

MATLAB

• A = [Ax Ay Az];• norm (A)

• A = [3 4 12];• a = norm (A);• a• 13

A+B

A

B

A-B

A

B

-B

zyx uuuA 543

zyx uuuB 321

zyx uuuBA 864

zyx uuuBA 222

• a = 1ux + 2uy + 3uz

• b = 3ux + 2uy + 1uz

• c = a + b • c = 4ux + 4uy + 4uz

• >> a = [1 2 3];• >> b = [3 2 1];• >> c = a + b;• c =• 4 4 4

vectors - addition

cosABBA

B

A

ABBA

cosABBA

0if BAB

A

2

zyx uuuA 543

zyx uuuB 321

261583 BA

cosABBA

B

A

AB

zzz cosAA uA

x y

z

xx y

y

z

z A

MATLAB

• dot (A, B)

• A = [1 2 3];• B = [2 3 4];• c = dot (A, B);• c• 20

BAuBA sinAB

x y

z

AB

BAu

BAu

ABuAB sinBA

x y

z

A B

ABu

BAu

BAuBA sinAB

x y

z

0BAif

AB

πθ0θ

BAuBA sinAB

x y

z A

B

x y

z A

B

BAuBA sinAB

x y

z A

B

BAuBA sinAB

x y

z A

B

area = AB sin

BAuBA sinAB

xuA yuB

xu00 yu00 zu01

010001

zyx uuuBA

MATLAB• A = [1 0 0];• B = [0 1 0];

• c = cross (A, B);• c• 0 0 1• d = cross (B, A);• d• 0 0 -1

intersection of 3 planes

x

y

z

zyx uuuA zyx AAA

zyx uuuB zyx BBB

zzyyxx BABABA BA

zyx uuuA zyx AAA

zyx uuuB zyx BBB

zyx

zyx

zyx

BBBAAAuuu

BA

intersection of 2 planes

and a cylinder

x

y

z

z

r

There are MATLAB commands that permit you

to go back and forth between different

coordinate systems.

They are listed in the text for this class.

h2 d

h1 d

h3 d

11dh

2211 dhdh

332211 dhdhdh