vectors in a plane lesson 10.4. definitions vector: determined by direction and magnitude polar...
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Vectors in a PlaneLesson 10.4
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Definitions
• Vector: determined by direction and magnitude
• Polar representation: nonnegative magnitude r and direction θ measured counterclockwise from the polar axis is [r, θ]
• Standard position: initial point starts at the origin
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Example 1• A storm is traveling at 30 miles per hour in a
direction 15˚ north of east.• a. Draw an arrow in standard position representing the
velocity.
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Example 1• A storm is traveling at 30 miles per hour in a
direction 15˚ north of east.• b. Write the polar representation of the velocity vector
of the storm.[ 30, 15˚]
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Example 1• A storm is traveling at 30 miles per hour in a
direction 15˚ north of east.c. Describe the storm’s movement each hour in terms
of a number of miles east-west and a number of miles north-south.
East-west (green) = 30 cos 15Moving 29 miles east per hourNorth-south(red) = 30 sin 15Moving 7.8 miles north per hour
30
15˚
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Component Representation• Component representation of a plane
vector u is the ordered pair, < u1, u2 >,
the rectangular coordinates of the point at the end of the standard position arrow.
Horizontal component Vertical
component
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Example 2• Consider the vector represented by an arrow
from (60, 130) to (40, 290).• a. Find the length and direction of the vector.
2.16126000)130290()6040( 22 r
87.82)6040130290(tan 1 1.97180
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Back
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Example 2• Consider the vector represented by an arrow from (60, 130) to (40, 290).• b. In standard position, what is the ending point of the vector?
(40 – 60, 290 – 130) = (-20, 160)convert [ 161.2, 97.1] to rectangular
92.191.97cos2.161 x96.1591.97sin2.161 y
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Norm of a vector• If u = <u1, u2> then
• Zero Vector: (0,0) length 0 in any direction• Unit vectors:
<1,0> = i<0,1>= j
22
21|||| uuu
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Example 3
• A person is exerting a force of 80 pounds in a direction 22˚ east of north to move a sofa. What are the x and y components of the force vector?
• [80, 68˚]• x = 80cos 68 = 29.9685• y = 80sin 68 = 74.174
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Homework
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