Vectors in a Plane Lesson 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 15 Moving 29 miles east per hour North-south(red) = 30 sin 15 Moving 7.8 miles north per hour 30 15˚

Component Representation Component representation of a plane vector u is the ordered pair,, 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).represented a. Find the length and direction of the vector.

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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

Norm of a vector If u = then Zero Vector: (0,0)  length 0 in any direction Unit vectors: = i = j

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 = y = 80sin 68 =

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