vector intro

8/2/2019 Vector Intro

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Introduction to Vectors

Definition Vector:

Examples Non-Examples1.

2.3.

Vectors have an initial point (where it starts), and a terminal point(where it ends).

For example, imagine a molecule moving from a point A to a point B,this movement could be modeled by a displacement vector where itslength represents how far it moved, and it would point in its direction ofmovement.

Two vectors are considered to be equal if:1.2.

The zero vector is denoted 0, has length 0 and has no particular direction.

Magnitude and Direction in Standard PositionFinding magnitude and direction of vector in standard position:

1. To find the magnitude of a vector not in the coordinate plane, just

measure it with a ruler.

2. To find the direction of a vector in standard position that is not in thecoordinate plane, use a protractor to the measure the degree of rotationfrom the positive x-axis. Just like angles in trigonometry.

Find the magnitude and direction of the following vectors.a. b.

x

v

A

B


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Combining Vectors GeometricallyGeometric definition of vector addition:Ifu and v are vectors positioned so the initial point ofv is at the terminal pointofu, then the sum u + v is the vector from the initial point ofu to the terminalpoint ofv.

Triangle Law:

Parallelogram Law: (u + v = v + u)

We will investigate three types of vector multiplication. First we will look atscalar multiplication.

Ex1: Take vectors u and v below and combine them the following ways. Drawthe resultant vector, and find the magnitude and direction of the resultantvector.

a. u + v b. u v

c. 2v d. 2u v

e. )( vv +

u v


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Component Form of Vectors

Vectors can also be represented by an ordered set of numbers and can besketched on a coordinate system with initial point at the origin and terminal

point at ( )21,aa

A 2-D vector can be given in component form as 21,aa

Remember that any vector that has the same length as u and is parallel to u isan equivalent vector.

Ex1: Find the vector AB where A(-2, 3) and B(1, 5). Graph AB in standard

position and find || AB . Lastly, find the direction ofAB

The Magnitude or Length of a vectora = 21,aa can be calculated using the distance

formula based on the Pythagorean Theorem =a _______________

A displacement vector from ),( 11 yxA to ( )22 , yxB given by

,=AB

The reference angle of a vectora = 21,aa can be found by :

Remember, that you still need to find the direction depending on whatquadrant the vector is in


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In order to add vectors in component form, add the respective components.

Ex2. Let u = 2,3 and v = 2,4 . Calculate:

a. vu + b. vu

b. v2 c. vu 34

Standard Unit Vectors:

i = 0,1 j = 1,0 are special vectors called the standard unit vectors. They

have length 1 (unit) and point in the direction of the x and y-axis respectively.

Any vector written in component form can be rewritten as the sum of amultiple of these two vectors.

Ex 3: Rewrite 9,6 in terms of the standard unit vectors

Ex 4: Find a unit vector parallel in the direction of 8,6

Ex 5: Using the vector diagram below, find the following for each vector: (a) the direction,

(b) the magnitude, (c) the vector in unit vector form, and (d) the unit vector in the samedirection as the given vector.

5, 3a = r

1, 4w = r

3, 5v = r

3, 5u =r

x

y


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vector intro

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