Vectors in a PlaneLesson 10.4

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

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.

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˚]

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˚

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

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

Back

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

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

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

Homework

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