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Properties and Components of Vectors
Physics 101Monday February 3, 2020
Dr. Adam Kobelski
Homework For Wednesday:Problems 3.001,3.010, 3.013Enter questions on webAssign
Exam is Wednesday 7-10 pm
• 20 Multiple Choice Questions• Dr. Holcomb’s Class (11:30-12:20)• Right Here B51.
• Dr. Kobelski’s Class• Eiesland Hall G24
• Accommodations• White Hall G04
ReviewScalars vs Vectors
Problem Solving Strategy
Displacement
Velocity
Acceleration
1D motion (kinematic equations)
Falling!
2D vectors (today)
A ball is tossed straight up into the air. Which of the following represents the signs of the acceleration as it moves upward, reaches its highest point, and falls back down? Moving upward Highest point Falling back
down
A. + 0 -
B. - 0 -
C. - - -
D. None of these
Q01
Don’t Share with Your Neighbor y
- 0 +
E. None of these
Define up to be the positive y direction.
Main Ideas in Class Today
After today, you should be able to:• Understand vector notation• Use basic trigonometry in order to find the x
and y components of a vector (only right triangles)
• Add and subtract vectors
Practice Problems: 3.3, 3.5, 3.7, 3.9, 3.11, 3.15, 3.17, 3.19, 3.21
Quick ReviewQuantities that are determined by a magnitude alone are called scalars.Quantities that have both magnitude and direction are called vectors.
e.g. displacement, velocity, acceleration
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
+xEast
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
Describing a vector
A vector is described *completely* by two quantities:magnitude
(How long is the arrow?)&
direction(What direction is the arrow pointing?)
+y
+x
Magnitude and direction
North+y
+xEast
50 miles East
30 miles North
Magnitude:
length of this li
neDirection:angle from reference point (here, “θ degrees North of East”)θ
Vector notation+y
+x
c
d
This vector written down:
And its magnitude…~A A cd
| ~A| |A| |cd|
Vector “components”
+y
+x
Ax
Ay
“Vector change in y direction”
“Vector change in x direction”
~Ay
~Ax
Vector Direction
North+y
+xEast
A car drives 4 miles east and 3 miles north. What is the direction of the displacement from its starting point?
Displacement is a vector(net change in position)
4 miles East
3 miles North
A) NEB) NWC) SED) SWE) E Q02
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.
Q03
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
Step 1: TranslateStep 2: Add tail to tipThis 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?
Q04
E. None of the above
A. B.
C. D.
Step 1: Translate the vector Step 2: Add tail to tip!
V1 + V2 = VR
Vector Addition
What is
E. None of the above
A. B.
C. D.
+ = ? Q05
Basic vector operations
+y
+x
Geometrically subtracting vectors
When adding/subtracting geometrically, always add tail to tip!
~A� ~B =?
-
+y
+x
Basic vector operations
Geometrically subtracting vectors
When adding/subtracting geometrically, always add tail to tip!
~A� ~B = ~A+⇣� ~B
⌘
-
+y
+x
Basic vector operations
Geometrically subtracting vectors
~A� ~B = ~A+⇣� ~B
⌘
When adding/subtracting geometrically, always add tail to tip!
Basic vector operations
North+y
+xEast
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
Addition and subtraction
North+y
+xEast
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
Basic vector operationsAddition and subtraction
Fig. 3.4 in your book
R
Basic vector operationsAddition and subtraction
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.
A hiker goes on a 2-day hike. On the first day, the hiker travels 25 km Northeast. On day 2, the hiker travels 30 km East. Find the total displacement (magnitude and direction) from the point of origin.
The total amount that you go East, is the amount you go East on Day 1 plus the amount that you go East on Day 2.
What would I do if I backtracked some?
Basic vector operationsScalar multiplication
3 -3
Multiplying a vector A and a scalar (i.e. number) kmakes a vector, denoted by kA.
North+y
+xEast
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
Q06 Think about the Pythagorean Theorem
Vector arithmetic: components
Vector arithmetic: components
AAy
Ax xfxi
yf
yi
|~Ay| =Ay
=
yf�yi
|~Ax| =Ax = xf � xi
Addition: 21 VVV!!!
+=
xxx VVV 21 +=
yyy VVV 21 +=
• When adding vectors, components are added separately
• Never add magnitudes of vectors
Vector Arithmetic – Components(Will be important in Chapter 4)
Vector arithmetic: components
AAy
Ax xfxi
yf
yi
We don’t always have the coordinates though. We can use our trig functions!
|~Ax| = A cos ✓ = AAx
A= Ax
|~Ay| = A sin ✓
= AAy
A= Ay
