Vectors!!!

Adding Vectors graphically

A

B

Tail-to-tip method

A

B

A

B

C

C = A + B

Vectors: By components

This is big, the next ~7 chapters use this

We will introduce it today, finishing the chapter

and then jump into application of them on

Wednesday.

This will free Friday for an activity, it would be good to bring your computers on Friday.

Y

X

V

V

V

y

x

The idea is that every vector has

components (parts) that are in the

direction of the x and y axis.

Vectors each have a magnitude and a

direction.

Knowing these two thing define a vector.

But there is another way to define a vector

If we know its vector components it is also

defined.

O

Now for a bit of Trig review

That's nice how do I use it?

What do I NEED to make

use of it?

Two things.

Right triangle

Angle

O

O

Hypotenuse

Opposite side

Adjacent side

O

5

4

3

A. 3/4 B. 3/5 C. 4/5 D. 4/3 E.5/4

What is the cosine of theta?

What is the tangent of theta?

V

y

V

y

V

x

V

x

V

V

V

V

Sin =

Tan =

= Sin

= Cos

O

O

O

O

V

x

V

Cos =

O

y

A

B

B

A

x

x

Y

Y

B

A

x

x

A

Y

C

x

C

Y

C

B

Y

How many of these quantities do you need to define a vector?

A. 1 B. 2. C. 3 D. 4 E. depends

**Unit vector**

These are as they are defined

Vectors of length one, unity.

They depend on the coordinate system,

and are very useful for the transformation

between coordinate systems as well as

analysis of questions.

Standard notation for these are:

in the x direction

in the y direction

in the z direction

X

Y

Z

With these unit vectors we can define our

vectors and component vectors as:

**Let vectors**

A = (3.0 m, 20 degrees south of east),

B = (2.0 m, north)

C = (5.0 m , 70 degrees south of west)

Write A,B,C in component form, and find D=A+B+C.

A = (3.0 m, 20 degrees south of east),

B = (2.0 m, north)

C = (5.0 m , 70 degrees south of west)

Write A,B,C in component form, and find D=A+B+C.

**3.9 m 73 degrees south of east**

**Let**

find the magnitudes and directions of:

E and F

E + F

-E - 2F

find the magnitudes and directions of:

E and F

E + F

-E - 2F

3.6, 56.3 degrees north of east

2.8, 45 degrees south of east

4.1, 14 degrees north of east

6.1, 9.5 degrees north of west

**Which of these two are easier? Why.**

As we do problems in the future, we will

use these two ideas heavily

Practice, Practice Practice

3. 10,11,12,13,14,15,16,21,23,29

As we do problems in the future, we will

use these two ideas heavily

Practice, Practice Practice

3. 10,11,12,13,14,15,16,21,23,29

**Draw each of the following vectors, label an angle that specifies the vector's direction, then find its magnitude and direction.**

**Write F in component form**

Draw all three

What are the magnitude and direction of F?

Draw all three

What are the magnitude and direction of F?

**23 For the three vectors shown:**

What is vector B, in components form and magnitude and direction?

What is vector B, in components form and magnitude and direction?

**C 2**

**4 A**

**B**

"Could you go over the types of coordinate systems; polar and Cartesian?"

"Can we go over the components of the vectors just a little bit?" "Can you please explain the difference between components and component vectors?"

"why does it say that the resultant vector is the sum of the x and y component vectors when we actually need to use the Pythagorean theorem to find the resultant vector?"

"Are vector components and a vector's magnitude and direction the same thing? Can there be a circumstance where a vector would be decomposed into more than 2 composite vectors?"

"Will you explain a little more on how to use the coordinate systems?"