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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
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