February 25, 2015
Section 6.1
Vectors in the Plane
A two-dimensional vector v is an ordered
pair of real numbers, denoted in component
form as
. The numbers a and b are the
components of the vector v.
The standard representation of the vector
is an arrow from the origin to the point (a, b).
The magnitude of v is the length of the arrow,
and the direction of v is the angle form by the
arrow with the positive x-axis
The vector 0 =
called the zero vector,
has zero length and no direction.
Equivalent vectors
Q(3, 5)
S(7, 2)
To show that to vectors
equivalent we need to show
that they have the same
length and direction. How
can we do this?
P(0, 0)
R(4, -3)
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The magnitude or length of the vector
determined by
The sum of the vectors
is the vector
The product of the scalar k and the vector
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θ
Find the components
of a vector when
given the magnitude
and direction.
Find the component form of the vector v
with magnitude 10 and direction angle 135°.
provided you
convert θ to an angle measure in standard
position.
Find the magnitude and direction angle of
the vector
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unit vector.
is a unit vector in the direction of
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The velocity of a moving object is a vector because it has
both magnitude and direction. The magnitude of velocity
is speed. A bearing is the angle that the line of travel
makes with due north, measured clockwise.
Find θ, the angle in standard
position, if the bearing is 205°.
42. An airplane is flying on a bearing of 170° at 460 mph.
Find the component form of the velocity of the airplane.
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An airplane is flying on a compass heading
(bearing) of 155° at 450 mph. A wind is
blowing with a bearing of 320° at 40 mph.
a) Find the component form of the
velocity of the airplane.
p = 〈450cos295°, 450sin295°〉
b) Find the component form of the wind.
w = 〈40cos130°, 40sin130°〉
c) Find the actual ground speed and direction of the plane.
v
= 〈450cos295°+40cos130°, 450sin295°+40sin130°〉
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