vector mathematics physics 1 physical quantities a scalar quantity is expressed in terms of...
TRANSCRIPT
![Page 1: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/1.jpg)
VectorMathematics
Physics 1
![Page 2: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/2.jpg)
Physical Quantities
A scalar quantity is expressed in terms of magnitude (amount) only.
Common examples include time, mass, volume, and temperature.
![Page 3: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/3.jpg)
Physical Quantities
A vector quantity is expressed in terms of both magnitude and direction.
Common examples include velocity, weight (force), and acceleration.
![Page 4: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/4.jpg)
Representing Vectors
Vector quantities can be graphically represented using arrows.– magnitude = length of the arrow– direction = arrowhead
![Page 5: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/5.jpg)
Vectors
All vectors have a head and a tail.
![Page 6: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/6.jpg)
Vector Addition
Vector quantities are added graphically by placing them head-to-tail.
![Page 7: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/7.jpg)
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.
![Page 8: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/8.jpg)
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.
![Page 9: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/9.jpg)
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
![Page 10: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/10.jpg)
South
East
Resultant
Vector addition is ‘commutative’ (can add vectors in either order)
Resultant is (sqrt(2)) 45◦ south
of East
![Page 11: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/11.jpg)
South
East
Resultant
Vector addition is ‘commutative’ (can add vectors in either order)
Resultant is (sqrt(2)) 45◦ south
of East
South
East
Resultant
![Page 12: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/12.jpg)
Co-linear vectors make a longer (or shorter) vector
Resultant is 3 magnitude South
![Page 13: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/13.jpg)
Co-linear vectors make a longer (or shorter) vector
Resultant is 3 magnitude South
![Page 14: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/14.jpg)
Nor
thN
orth
EastEast
Can add multiple vectors.Just draw ‘head to tail’ for each vector
Resultant is magnitude
45◦ North of East 2
![Page 15: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/15.jpg)
Nor
thSo
uth
EastEast
Adding vectors is commutative.
Resultant is magnitude
45◦ North of East
Nor
th
East
Nor
th
East
22
![Page 16: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/16.jpg)
Nor
th
South
East
WestResultant=0
Equal but opposite vectors cancel each other out
Resultant is 0.
![Page 17: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/17.jpg)
Vector Addition – same directionA + B = R
B
A
A B
R = A + B
![Page 18: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/18.jpg)
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?
![Page 19: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/19.jpg)
Vector Addition – Opposite direction(Vector Subtraction) .
A + (-B) = RA
B
-B
A
-B A + (-B) = R
![Page 20: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/20.jpg)
Vectors• The sum of two or more vectors is called the
resultant.
![Page 21: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/21.jpg)
Practice
Vector Simulator http://phet.colorado.edu/sims/vector-addition/vector-addition_en.html
![Page 22: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/22.jpg)
Polar Vectors
Every vector has a magnitude and direction
direction anglemagnitude
![Page 23: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/23.jpg)
Right Triangles
SOH CAH TOA
![Page 24: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/24.jpg)
Vector Resolution
Every vector quantity can be resolved into perpendicular components.
Rectilinear (component) form of vector:
yx
![Page 25: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/25.jpg)
A
Ax
Ay
Vector Resolution
Vector A has been resolved into two perpendicular components, Ax (horizontal component) and Ay (vertical component).
![Page 26: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/26.jpg)
Vector Resolution
If these two components were added together, the resultant would be equal to vector A.
A
Ax
Ay
![Page 27: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/27.jpg)
Vector Resolution
When 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.
![Page 28: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/28.jpg)
Vector Resolution
Horizontal Component (Ax)
A
Axcos
cosAAx
A
Ax
Ay
![Page 29: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/29.jpg)
Vector Resolution
Vertical Component (Ay)
sin yA
A
sinyA A
A
Ax
Ay
![Page 30: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/30.jpg)
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
![Page 31: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/31.jpg)
Vector Resolution
Use the sign conventions for the x-y coordinate system to determine the direction of each component.
(+,+)(-,+)
(-,-) (+,-)
N
E
S
W
![Page 32: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/32.jpg)
Component Method
1. 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.
![Page 33: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/33.jpg)
Component Method
4. 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 q.
![Page 34: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/34.jpg)
Adding “Oblique” Vectors
Head to tail method works, but makes it very difficult to ‘understand’ the resultant vector
![Page 35: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/35.jpg)
Adding “Oblique” Vectors
Break each vector into horizontal and vertical components.
Add co-linear vectors Add resultant horizontal
and vertical components
![Page 36: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/36.jpg)
Adding “Oblique” Vectors
Break each vector into horizontal and vertical components.
![Page 37: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/37.jpg)
Adding “Oblique” Vectors
Break each vector into horizontal and vertical components.
Add co-linear vectors
![Page 38: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/38.jpg)
Adding “Oblique” Vectors
Break each vector into horizontal and vertical components.
Add co-linear vectors Add resultant horizontal
and vertical components
![Page 39: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/39.jpg)
Adding “Oblique” Vectors
Break each vector into horizontal and vertical components.
Add co-linear vectors Add resultant horizontal
and vertical components
![Page 40: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/40.jpg)
Using Calculator For Vectors
Can use the “Angle” button on TI-84 calculator to do vector mathematics
![Page 41: Vector Mathematics Physics 1 Physical Quantities A scalar quantity is expressed in terms of magnitude (amount) only. Common examples include time, mass,](https://reader036.vdocuments.us/reader036/viewer/2022062516/56649e6f5503460f94b6c5a3/html5/thumbnails/41.jpg)
Using Calculator for Vectors