February 22, 2010

6.1 Vectors in the Plane

vector: directed line segment, indicates magnitude and direction

P

QPQ

initial point

terminal pointv =

magnitude (length) = PQ

direction: the direction in which the arrow is pointing

equivalent vectors have the same magnitude and direction

February 22, 2010

standard position: initial point is at the origin

(a,b)

(0,0)

v

component form: v = <a,b>

zero vector: 0 = <0,0> (initial and terminal pts. at the origin) v =0

February 22, 2010

HMT Rule (Head Minus Tail Rule):

If an arrow has initial point (x1, y1) and terminal point

(x2, y2), it represents the vector x2- x2, y2-y1

1. Show u = v.

u is dir. line seg. from P(0,0) to Q(3,2)v is dir. line seg. from R(1,2) to S(4,4)

February 22, 2010

Do Exploration 1 on Pg. 504

February 22, 2010

2. Find the component form of vector v with initial pt. (4,-7) and terminal pt. (-1,5). Then find its magnitude.

February 22, 2010

scalar multiplication:

ku = k<u1,u2> = <ku1,ku2>

u = 3, 4

Find:

a. 2u

b. -u

c. 1 2

u

February 22, 2010

vector addition:

u

v

u+v

u + v = <u1+v1, u2+v2>

u + v is the resultant vector

u

v

u+v

Parallelogram Representation

Tail-to-Head Representation

February 22, 2010

negative of v is <-v1, -v2>

difference of u - v = <u1-v1, u2-v2>

u

v

u-v

u - v = u + (-v)-v

u + -v

February 22, 2010

3. v = <-2,5>, w = <3,4>

a. 2v b. w - v c. v + 2w

February 22, 2010

u = unit vector = v = 1 v v

v

unit vector in the direction of v.

February 22, 2010

4. Find a unit vector in direction of v = <-2,5> and verify the result has length of 1.

February 22, 2010

standard unit vectors

v = <v1, v2> = v1<1,0> + v2<0,1> = v1i + v2j

j = <0,1>

i = <1,0>

hor. comp vert. comp

February 22, 2010

5. u has initial pt. (2, -5) and term. pt. (-1,3). Write u as a linear combination of standard vectors i and j.

February 22, 2010

Direction Angles

unit vector

u

(x,y)

u = <x,y> = <cos O, sin O>

= (cos O)i + (sin O)j

direction angle = tan O = sin O cos O

Resolving the Vector:

If v has direction angle θ, the components of v can be computed using the formula:

v = < v cos θ, v sin θ >

February 22, 2010

Find the components of the vector v with direction

angle 120o and magnitude 5.

February 22, 2010

6. Find the direction angle

u = 3i + 3j

7. v = 3i - 4j

February 22, 2010

Velocity: A vector can represent velocity because it has

both magnitude and direction. The magnitude of velocity is speed.

An airplane is flying at a bearing of 170o at 460 mph. Find the component form of the velocity of the airplane.