Vectors  for    Physics  

AP  Physics  C  

A Vector … •  … is a quantity that has a magnitude (size)

AND a direction.

•  …can be in one-dimension, two-dimensions, or even three-dimensions

•  …can be represented using a magnitude and an angle measured from a specified reference

•  …can also be represented using unit vectors

Vectors in Physics

•  We only used two dimensional vectors •  All vectors were in the x-y plane. •  All vectors were shown by stating a

magnitude and a direction (angle from a reference point).

•  Vectors could be resolved into x- & y-components using right triangle trigonometry (sin, cos, tan)

Unit Vectors •  A unit vector is a vector that has a magnitude

of 1 unit

•  Some unit vectors have been defined in standard directions.

•  +x direction specified by unit vector “i” •  +y direction specified by “j” •  +z direction specified by “k” •  “n” specifies a vector normal to a surface

Using Unit Vectors

kji ˆ8ˆ5ˆ3 +−For Example: the vector

is three dimensional, so it has components in the x, y, and z directions.

The magnitudes of the components are as follows: x-component = +3, y-component = -5, and z-component = +8

The hat shows that this is a unit vector, not a variable.

Finding the Magnitude

( ) ( ) ( )222zyx AAAA ++=

To find the magnitude for the vector in the previous example simply apply the distance formula…just like for 2-D vectors

Where: Ax = magnitude of the x-component,

Ay = magnitude of the y-component,

Az = magnitude of the z-component

Finding the Magnitude

So for the example given the magnitude is:

( ) ( ) ( ) 899.9853 222 =+−+What about the direction?

In Physics we could represent the direction using a single angle measured from the +x axis…but that was only a 2D vector. Now we would need two angles, 1 from the +x axis and the other from the xy plane. This is not practical so we use the i, j, k, format to express an answer as a vector.

Vector Addition

kBjBiBB

kAjAiAA

zyx

zyx

ˆˆˆ

ˆˆˆ

++=

++=!

!

ˆˆ ˆ( ) ( ) ( )x x y y z zA B A B i A B j A B k+ = + + + + +r r

If you define vectors A and B as:

Then:

Example of Vector Addition

kjiA ˆ8ˆ5ˆ3 +−+=!

ˆˆ ˆ(3 2) ( 5 4) (8 ( 7))A B i j k+ = + + − + + + −r r

kjiB ˆ7ˆ4ˆ2 −++=!

ˆˆ ˆ5 1 1A B i j k+ = − +r r

If you define vectors A and B as:

Note: Answer is vector!

How many combinations of components can a vector have?

What’s happening here?

Vector Multiplication

! Also known as a scalar product.

! Measure of dependency of A and B

! Mag of A and component of B parallel to A are multiplied

BA!!

• BA!!

×Dot Product Cross Product

"  Also known as a vector product.

"  Measure of independency of A and A

"  Mag of A and component of B perpendicular to A are multiplied

Finding a Dot Product

kBjBiBB

kAjAiAA

zyx

zyx

ˆˆˆ

ˆˆˆ

++=

++=!

!

zzyyxx BABABABA ++=•!!

If you define vectors A and B as:

Where

Ax and Bx are the x-components, Ay and By are the y-components, Az and Bz are the z-components.

Then:

Answer is a Scalar only, no i, j, k unit vectors.

Example of Dot Product

kjiA ˆ8ˆ5ˆ3 +−+=!

( ) ( )784523 −∗+∗−+∗=•BA!!

kjiB ˆ7ˆ4ˆ2 −++=!

7056206 −=−−=•BA!!

If you define vectors A and B as:

Note: Answer is Scalar only!

Dot Products (another way)

θcosABBA =•!!

If you are given the original vectors using magnitudes and the angle between them you may calculate magnitude by another (simpler) method.

Where A & B are the magnitudes of the corresponding vectors and θ is the angle between them.

A

B

θ

Using  a  Dot  Product    in  Physics  

θcos∗∗= dFW

∫ •= dFW!!

Remember in Physics 1…To calculate “Work”

Where F is force, d is displacement, and θ is the angle between the two.

Now with calculus:

Note: This symbol means “anti-derivative”… we will learn this soon!

Dot product of 2 vector quantities

Right Hand Rule and Cross Product - What does it mean?

What is the direction of A X B?

Finding a Cross Product 3D

kBjBiBB

kAjAiAA

zyx

zyx

ˆˆˆ

ˆˆˆ

++=

++=!

!

BzByBxAzAyAxkji

BA

ˆˆˆ

=×!!

If you define vectors A and B as: Where

Ax and Bx are the x-components, Ay and By are the y-components, Az and Bz are the z-components.

Then:

Answer will be in vector (i, j, k) format.

Evaluate determinant for answer!

Find the determinants along with the sign of (-1) row#+Column#

Example of a Cross Product

kjiA ˆ8ˆ5ˆ3 +−+=!

kjiB ˆ7ˆ4ˆ2 −++=!

742853

ˆˆˆ

−

−=×

kjiBA!!

If you define vectors A and B as:

Set up the determinant as follows, then evaluate.

Evalua5ng  the  Determinant  

4253

ˆˆ

742853

ˆˆˆ

−

−

−=×

jikjiBA!!

)73(ˆ)48(ˆ)25(ˆ)43(ˆ)28(ˆ)75(ˆ −∗−∗−∗−−∗+∗+−∗−=× jikkjiBA!!

kjiBA ˆ)22(ˆ)37(ˆ)3( ++=×!!

Final answer in vector form.

Cross Products (another way)

θsinABBA =×!!

If you are given the original vectors using magnitudes and the angle between them you may calculate magnitude by another (simpler) method.

Where A & B are the magnitudes of the corresponding vectors and θ is the angle between them.

θ

B

A Note: the direction of the answer vector will always be perpendicular to the plane of the 2 original vectors. It can be found using a right-hand rule!

Using  a  Cross  Product    in  Physics  

τ = l ∗F ∗sinθ

τ = l ×!F

Remember in Physics 1…To calculate “Torque”

Where F is force, l is lever-arm, and θ is the angle between the two.

Cross product significance When will the Torque be more?

Some interesting facts

ABBA!!!!

•=•

( )ABBA!!!!

×−=×

The commutative property applies to dot products but not to cross products.

Doing a cross product in reverse order will give the same magnitude but the opposite direction!

Problems

110 deg

In the figure, vector a lies in the xy plane, has a magnitude of 18 units and points in a direction 250° from the positive direction of the x axis. Also, vector b has a magnitude of 12 units and points in the positive direction of the z axis. What is the vector product = a × b ?

216 @ 160 deg