Start Up Day 46

Vectors in the PlaneOBJECTIVE: SWBAT Represent vectors as directed line segments and write the component forms of vectors. SWBAT Perform basic vector operations and represent them graphically. SWBAT Write vectors as linear combinations of unit vectors and find the direction angles of vectors. . EQ: What is a vector? What 2 quantities are represented by a vector? What is a UNIT vector? HOME LEARNING:. p. 464 #2, 3, 5-8, 10, 14 – 20 even, 21 & 22+ Math XL Vector Review due (2/2)

A vector allows us to describe both a quantity and a direction of an object. If a physical quantity, such as speed, can be completely described by a single real number, this number is called a scalar. This number indicated the size or magnitude of the quantity. A vector is a quantity that has both magnitude and direction.Think of times when a vector might be used: the speed and direction of an airplane, the navigation of the path of a ship, or the path of a ball after being thrown or kicked.

http://www.youtube.com/watch?v=A05n32Bl0aY

1. A car traveling 40 mph.

2. A motorcycle traveling 60 mph due north.

3. A train traveling 40mph east to the beach.

4. A child’s weight on a scale.

1. A boat traveling 50 mph 20°east of north2. An object falling straight down at 15 mph3. A worker pushing an object with a force of 30

newtons

#3 There is a magnitude, but no direction!

Initial Point

or TAILTerminal Point

or HEADA

B

A vector is a quantity that has both magnitude (size of the quantity) and direction.The vector above can be written as a .The magnitude is written as When physical quantities can be represented by a single real number, they are called a scalar.

Vector Direction

αv

Standard Positionis measured counter clockwise from 0°.

vv

Quadrant bearing v is between 0° and 90° east or west of the north-south line. v is S 70°E

True Bearing is always measured clockwise from the north-south line. True bearings are always given in three digits. 050°

Vector Direction: Name the direction of vector r, in three different ways.

r Standard Position:

Quadrant Bearing:

True Bearing:

120°

N 30°W

330°

Vectors in two point form Given P (1, 2) and Q(4, 4), the vector PQ can be drawn in

the coordinate plane. P is assumed to be the initial point and Q is given as the

terminal point.Component Form of a vector is a way to rewrite the vector

in standard form. Direction and Length is maintained! Terminal –

Initial

Component form “moves” the initial point to the origin

QP

Magnitude of a vectorMagnitude is LENGTH!

To determine the length of the vector just call on Pythagoras!

To see the right triangle—just use “a” or the x- component as one leg and “b” or y-

component as the other.

EXAMPLE 2: Determine the component form and the

magnitude of the vector v Initial Point ( -3, 4) and Terminal Point ( -5 , 2).

First Rewrite in component form:

Next, determine the magnitude:

2,242),3(5

22822 22 v

Resultant?

EXAMPLE 3: VECTOR OPERATIONSalgebraically-- a bit intuitive!

Given Determine:a. u + v = b. 3 u = c. 2u-v=

10,37,43,1 9,33,13

1,67,46,27,43,12

7,4&3,1 vu

When two or more vectors are added, their sum is a single vector called the resultant.

One method for adding vectors is called the triangle method and is illustrated graphically below.

a

b

a b

a b

a+b

Think of the arrow as the “Head” and the initial point as the “Tail”.To determine the resultant

of a and b, translate the 2nd vector so that the tail of “b” starts from the head of “a”.

The resultant is the vector from the “Tail” of a to the “Head” of b. (It closes the triangle!)

Unit VectorsUnit Vectors bring us back to the “circle of life”

Unit Vectors are the cosine and sine of the directional angle and have a magnitude of “1”

To determine a unit vector in the same direction as a given vector, just divide each

component by its magnitude.The standard unit vectors are denoted by “i”

and “j”http://www.youtube.com/watch?feature=player_embedded&v=6o_S7u7Ddx4

Your Turn: Determine a unit vector in the same direction as the given

vector?V = 6i – 2 j = 6 - 2 =

First re-write in component form, then calculate magnitude,

Now, divide each component of your vector by its magnitude & then you will have your

UNIT vector!

1024026 22 v

Start Up Day 47

1. Determine the magnitude and directional angle of v: v = -4i -4j

2. Determine a UNIT vector in the same direction of v

Vectors in the Plane—Day 2Lesson 6.1

OBJECTIVE: SWBAT Represent vectors as directed line segments and write the component forms of vectors. SWBAT Perform basic vector operations and represent them graphically. SWBAT Write vectors as linear combinations of unit vectors and find the direction angles of vectors. . EQ: How do we use a UNIT vector to create a desired magnitude? How is velocity represented? How is speed represented?

Back to the UNIT Circle!

Directional Angle:

The directional angle can most easily be found by thinking back to SOHCAHTOA.

If you sketch your vector in component form, your “x” coordinate is your adjacent side and

your “y ” coordinate is your opposite side.Use the inverse tangent function—carefully.

(It’s only right if its on the right) If your vector is on the left—

just add 180 to your calculator answer

v

vv x

y1tan

https://youtu.be/fVq4_HhBK8Y

Example #8 cont.Example #8 cont.