warm upmay 9 th 1. in a pumpkin tossing contest in morton, illinois, a contestant won with a...

Warm Up May 9th 1. In a pumpkin tossing contest in Morton, Illinois, a contestant won

with a catapult that launches the pumpkin with an initial speed of 125 feet per second, at an angle of 50°, and from an initial height of 25 feet.

a) Write a set of parametric equations for the motion of the pumpkin.

b) How long was the pumpkin in the air? (2 decimal places)c) How far did the pumpkin travel? (nearest foot)

2. The pumpkin tossing contest was at a state fair where they had a Ferris wheel that was 205 feet high. It takes the Ferris wheel 2 minutes to complete 1 revolution.

a) If a rider starts at the point (0, 5) when t = 0 (in seconds) and moves clockwise, write a set of parametric equations to model the ride.

b) How far off the ground is the rider after 50 seconds?

Homework Check

Vectors

What is a vector?A directed line segment

initial point

P

Q

Terminal point

This is vector PQ

Vectors with the same length and direction are

equivalent.

Let PQ be a vector with P(1,2) and Q(4,7). Find the terminal point of the equivalent vector with initial point (0,0)

• A vector whose initial point is (0,0) is in standard position.

• In standard position the vector can be represented by the coordinates of the terminal point.

• This is called component form. <v1,v2>

• Vectors in standard position are named with one lower case bold letter. v

Find the component form of the vector with initial point (-2,3) and terminal point (4, -1).

The length of a vector is called its magnitude. It is denoted |v|.

• Given u = < -1, 5>. Find |u|.

• Given AB with A(2, -4) and B(-1, 0). Find |AB|.

The direction of the vector is the angle, , that the vector makes with the positive x-axis. (always give the angle as a positive number less than 360°)Find the direction of the vector 1) u = < -1, 5>

2) v = <3, -4>

3) AB with A(1, -4) and B(3, 5).

Vector operationsGiven u = <-2, 5> and v = <3,4>Findu + vu – v3u (this is called scalar multiplication)2v – 3u

What effect does scalar multiplication have on the

vector?

1. Find the vector in the direction 45° that has magnitude 1.

2. Find the vector in the direction 240° that has magnitude 4.

A unit vector is a vector with magnitude one.• Find the unit vector in the same direction

as <-3,4>.

• Find the unit vector in the same direction as <2, 5>

• Find the unit vector in the direction 30°

Linear Combination of VectorsSince i = <1, 0> and j = <0, 1> are the

standard unit vectors, v = <3, 4> can be written as 3i + 4j.

1) Rewrite <2,-5> using i and j.

2) Rewrite in component form: -3i + 8j

1) Find a vector that has magnitude 2 and is in the same direction as -6i – 8j

2) Find the vector that has magnitude 5 and the same direction as <3, -2>.

Let’s summarize…• How do you get the component form of PQ if P(a, b) and

Q(c, d)?

• How do you find the magnitude of v?

• How do you find the direction of v?

• How do you add or subtract vectors?

• What is a unit vector and how do you find it?

• How do you find the components of a vector given the direction and magnitude?

• How do you get a vector to be a different length?