6.1 vectors in the plane vector: directed line segment, indicates magnitude and direction ·...
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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
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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
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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)
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Do Exploration 1 on Pg. 504
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2. Find the component form of vector v with initial pt. (4,-7) and terminal pt. (-1,5). Then find its magnitude.
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scalar multiplication:
ku = k<u1,u2> = <ku1,ku2>
u = 3, 4
Find:
a. 2u
b. -u
c. 1 2
u
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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
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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
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3. v = <-2,5>, w = <3,4>
a. 2v b. w - v c. v + 2w
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u = unit vector = v = 1 v v
v
unit vector in the direction of v.
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4. Find a unit vector in direction of v = <-2,5> and verify the result has length of 1.
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standard unit vectors
v = <v1, v2> = v1<1,0> + v2<0,1> = v1i + v2j
j = <0,1>
i = <1,0>
hor. comp vert. comp
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5. u has initial pt. (2, -5) and term. pt. (-1,3). Write u as a linear combination of standard vectors i and j.
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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 >
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Find the components of the vector v with direction
angle 120o and magnitude 5.
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6. Find the direction angle
u = 3i + 3j
7. v = 3i - 4j
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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.