VectorMathematics

Physics 1

Physical Quantities

A scalar quantity is expressed in terms of magnitude (amount) only.

Common examples include time, mass, volume, and temperature.

Physical QuantitiesA vector quantity is expressed in

terms of both magnitude and direction.

Common examples include velocity, weight (force), and acceleration.

Representing VectorsVector quantities can be graphically

represented using arrows.– magnitude = length of the arrow– direction = arrowhead

Vectors All vectors have a head and a tail.

Vector Addition

Vector quantities are added graphically by placing them head-to-tail.

Head-to-Tail Method

1. Draw the first COMPONENT vector with the proper length and orientation.

2. Draw the second COMPONENT vector with the proper length and orientation starting from the head of the first component vector.

Head-to-Tail Method

3. The RESULTANT (sum) vector is drawn starting at the tail of the first component vector and terminating at the head of the second component vector.

4. Measure the length and orientation of the resultant vector.

South

East

Resultant

To add vectors, move tail to head and then draw resultant from original start to final point.

Resultant is (sqrt(2)) 45◦ south

of East

South

East

Resultant

Vector addition is ‘commutative’ (can add vectors in either order)

Resultant is (sqrt(2)) 45◦ south

of East

South

East

Resultant

Vector addition is ‘commutative’ (can add vectors in either order)

Resultant is (sqrt(2)) 45◦ south

of East

South

East

Resultant

Co-linear vectors make a longer (or shorter) vector

Resultant is 3 magnitude South

Co-linear vectors make a longer (or shorter) vector

Resultant is 3 magnitude South

Nor

thN

orth

EastEast

Can add multiple vectors.Just draw ‘head to tail’ for each vector

Resultant is magnitude

45◦ North of East 2

Nor

thSo

uth

EastEast

Adding vectors is commutative.

Resultant is magnitude

45◦ North of East

Nor

th

East

Nor

th

East

22

Nor

th

South

East

WestResultant=0

Equal but opposite vectors cancel each other out

Resultant is 0.

Vector Addition – same directionA + B = R

B

A

A B

R = A + B

Vector Addition

• Example: What is the resultant vector of an object if it moved 5 m east, 5 m south, 5 m west and 5 m north?

Vector Addition – Opposite direction(Vector Subtraction) .

A + (-B) = RA

B

-B

A

-B A + (-B) = R

Vectors• The sum of two or more vectors is called the

resultant.

Practice

Vector Simulator http://phet.colorado.edu/sims/vector-addition/vector-addition_en.html

Polar VectorsEvery vector has a magnitude and

direction

direction anglemagnitude

Right Triangles

SOH CAH TOA

Vector ResolutionEvery vector quantity can be

resolved into perpendicular components.

Rectilinear (component) form of vector:

yx

A

Ax

Ay

Vector Resolution

Vector A has been resolved into two perpendicular components, Ax (horizontal component) and Ay (vertical component).

Vector ResolutionIf these two components were

added together, the resultant would be equal to vector A.

A

Ax

Ay

Vector ResolutionWhen resolving a vector graphically, first

construct the horizontal component (Ax). Then construct the vertical component (Ay).

Using right triangle trigonometry, expressions for determining the magnitude of each component can be derived.

Vector Resolution

Horizontal Component (Ax)

AAxcos

cosAAx

A

Ax

Ay

Vector Resolution

Vertical Component (Ay)

sin yAA

sinyA A

A

Ax

Ay

Drawing Directions

EX: 30° S of W– Start at west axis and move south 30 °– Degree is the angle between south and west

N

S

EW

Vector ResolutionUse the sign

conventions for the x-y coordinate system to determine the direction of each component.

(+,+)(-,+)

(-,-) (+,-)

N

E

S

W

Component Method1. Resolve all vectors into

horizontal and vertical components.

2. Find the sum of all horizontal components. Express as SX.

3. Find the sum of all vertical components. Express as SY.

Component Method4. Construct a vector diagram using the

component sums. The resultant of this sum is vector A + B.

5. Find the magnitude of the resultant vector A + B using the Pythagorean Theorem.

6. Find the direction of the resultant vector A + B using the tangent of an angle .

Adding “Oblique” Vectors

Head to tail method works, but makes it very difficult to ‘understand’ the resultant vector

Adding “Oblique” Vectors

Break each vector into horizontal and vertical components.

Add co-linear vectors Add resultant horizontal

and vertical components

Adding “Oblique” Vectors

Break each vector into horizontal and vertical components.

Adding “Oblique” Vectors

Break each vector into horizontal and vertical components.

Add co-linear vectors

Adding “Oblique” Vectors

Break each vector into horizontal and vertical components.

Add co-linear vectors Add resultant horizontal

and vertical components

Adding “Oblique” Vectors

Break each vector into horizontal and vertical components.

Add co-linear vectors Add resultant horizontal

and vertical components

Using Calculator For Vectors

Can use the “Angle” button on TI-84 calculator to do vector mathematics

Using Calculator for Vectors