VectorsVector Practice Problems:

Odd-numbered problems from 3.1 - 3.21

Reminder: Scalars and VectorsVector:

A number (magnitude) with a direction.

a +xv

I have continually asked you, “which way are the v and a vectors pointing?”

Scalar:Just a number.

Vectors

North +y

+x East

A car drives 50 miles east and 30 miles north. What is the displacement of the car from its starting point?

Displacement is a vector

(net change in position)

50 miles East

30 miles North

A vector is described *completely* by two quantities:magnitude

(How long is the arrow?)

direction(What direction is the arrow pointing?)

Describing a vector+y

+x

&

Magnitude and direction

North +y

+x East

50 miles East

30 miles North

Magnitude:

length of this lin

e Direction:angle from

reference point (here, “θ degrees

North of East”)θ

Vector notation+y

+x

c

d

This vector written down: cdAA

And its magnitude…|cd||A||A|

Vector “components”+y

+x

Ax

Ay

Ax

Ay

“Vector change in y direction”

“Vector change in x direction”

Basic vector operations

+y

+x

Vectors are defined by ONLY magnitude and direction.

= = =

These are all the SAME vector!

Translating vectors

Basic vector operations

+y

+x

–V, has an equal magnitude but opposite direction to V.

— =

Multiplying by -1

In which case does = ⎻ ?A. B.

C. D.

Q17

Basic vector operations

+y

+x

Two vectors with the SAME UNITS

can be added.

[m/s]

[m/s]

Geometrically adding vectors

Basic vector operations

+y

+x

+ = ?

Geometrically adding vectors

When adding geometrically, always add tail to tip!

tiptail

Basic vector operationsGeometrically adding vectors

+y

+x

+ = vector + vector = vector

This is called the “triangle method of addition”

Basic vector operationsGeometrically adding vectors

+y

+x

+ =

+ =

It’s commutative!It doesn’t matter which one you add first.

If you were to add these two vectors, roughly what direction

would your result point? Q18

E. None of the above

A. B.

C. D.

Translate the vector and always add tail to tip!

V1 + V2 = VR

What is Q19

E. None of the above

A. B.

C. D.

+ = ?

Basic vector operations

+y

+x

Geometrically subtracting vectors

When adding/subtracting geometrically, always add tail to tip!

- = ?

-

Basic vector operations

+y

+x

Geometrically subtracting vectors

- = + (- )

When adding/subtracting geometrically, always add tail to tip!

-

Basic vector operations

+y

+x

Geometrically subtracting vectors

- = + (- )

When adding/subtracting geometrically, always add tail to tip!

Vectors

North +y

+x East

A car drives 50 miles east and 30 miles north. What is the displacement of the car from its starting point?

50 miles East

30 miles North

Vectors

North +y

+x East

A car drives 50 miles east and 30 miles north, then 20 miles south. What is the displacement of the car from

its starting point?

20 miles South

In graphical addition/subtraction, the arrows should always follow on from one another, and the resultant vector should always go from the starting point to the destination point in your summed vector path.

Fig. 3.4 in your book

R

Basic vector operationsScalar multiplication

3 -3

Multiplying a vector A and a scalar (i.e. number) k makes a vector, denoted by kA.

Intermission

North +y

+x East

50 miles East

30 miles North

Dx

Dy

What is D (the magnitude of )?

A. 58 milesB. 80 milesC. 20 milesD. 0 milesE. 58 m/s

Q20 Think about the Pythagorean Theorem

Vector arithmetic: components

Vector arithmetic: components

A, Ay, and Ax here are the MAGNITUDES of the vectors drawn

(they don’t have hats and are not bold).

AAy

Ax

Vector arithmetic: components

The magnitude of a vector component is its final number

minus initial number!

AAy

Ax

Vector arithmetic: components

|Ax| = Ax = xf - xi

AAy

Ax

|Ay| = Ay

= yf - yi

xfxi

yf

yi

What is the magnitude of the x component of ?

North +y

+x East

Dx

DyA. 60.6 milesB. 35.0 milesC. 40.4 milesD. 0 milesE. 31 degrees

Q21 Think about SOH CAH TOA!

70 miles

30o

Vector arithmetic: components

D

D

Using trigonometry,you can find all vector components and

angles given just a bit of information!

Look at the triangles, and think about what you can figure out based on available info.

Ultimate rule of vector mathDon’t fear the vector.

To study:Practice drawing/graphing vector operations.Get used to vector and magnitude notations.

Practice solving for x, y components.Practice solving for θ.

TO START:Draw your vector right triangle.

What are the sides? Compare your triangle with your neighbor.

Diandra kicks a soccer ball to a max height of 5.4 m at a 20° angle from the ground with a speed of 30 m/s. What is the x (horizontal) component of

the initial velocity of the soccer ball?

30 m/s

20°

5.4 m

CAREFUL! You can’t do vector arithmetic combining displacement

(5.4m) with speed (30m/s)!+y

+x