11.1 VECTORS IN THE PLANE MISS BATTAGLIA - AP CALCULUS OBJECTIVE: DETERMINE POSITION, VELOCITY, AND...

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COMPONENT FORM OF A VECTOR

Quantities (such as force, velocity, and acceleration) involve magnitude and direction. A directed line segment is used to represent these quantities. The directed line segment has an initial point P and terminal point Q, and its length (or magnitude) is denoted .

COMPONENT FORM OF A VECTOR

Directed line segments that have the same length and direction are equivalent. The set of all directed line segments that are equivalent to a given directed line segment is a vector in the plane.

VECTOR REPRESENTATION BY DIRECTED LINE SEGMENTSLet v be represented by the directed line segment

from (0,0) to (3,2) and let u be represented by the directed line segment from (1,2) to (4,4). Show that v and u are equivalent.

AN INTRODUCTION TO VECTORS

If a vector starts at ( x1, y1 ) and terminates at ( x2, y2 ), then its components are

< x2 – x1, y2 – y1 >

The magnitude is the length of the vector.

Find the component form for each vector. Find the magnitude of the vector.

a.Initial point of (2, 3) and terminal point of (7, 6)

b.Initial point of (3, 1) and terminal point of (2, - 3)

DEFINITIONS OF VECTOR ADDITION AND SCALAR MULTIPLICATION

Let and be vectors and let c be a scalar.

1.The vector sum of u and v is the vector

2.The scalar multiple of c and u is the vector

3.The negative of v is the vector

4.The difference of u and v is

VECTOR OPERATIONS

Given and , find each of the vectors.

a. ½v

b. w – v

c. v + 2w

PROPERTIES OF VECTOR OPERATIONS

Let u, v, and w be vectors in the plane and let c and d be scalars.

1. u + v = v + w

2. (u + v) + w = u + (v + w)

3. u + 0 = u

4. u + (-u) = 0

5. c(du) = (cd)u

6. (c + d)u = cu + du

7. c(u + v) = cu + cv

8. l(u) = u, 0(u) = 0

A car travels with a velocity vector given by:

where t is measured in seconds, and the vector components are measured in feet.If the initial position of the car is:

find the position of the car after 1 second.