1. Lecture 4: Vectors & Components

2. Questions of Yesterday
1) A skydiver jumps out of a hovering helicopter and a few seconds later a second skydiver jumps out so they both fall along the same vertical line relative to the helicopter.
1a) Does the difference in their velocities:
a) increase
b) decrease
c) stay the same
1b) What about the vertical distance between them?
2) I drop ball A and it hits the ground at t1. I throw ball B horizontally (v0y = 0) and it hits the ground at t2. Which is correct?
a) t1 < t2
b) t1 > t2
c) t1 = t2

3. Vector vs. Scalar Quantities
Vector Quantities: Magnitude and Direction
Ex. Displacement, Velocity, Acceleration
Scalar Quantities: Magnitude
Ex. Speed, Distance, Time, Mass
What about 2 Dimensions?
Vectors in 1 Dimension
Direction specified
solely by + or -
x (m)
y (m) 1 2 3 -3 -2 -1
x (m) 1 2 3 -3 -2 -1

4. Vectors: Graphical Representation
Vector Quantities: Magnitude and Direction
Represent in 2D with arrow
Length of arrow = vector magnitude
Angle of arrow = vector direction
y (m)
R m at qo above x-axis 3
Position of vector
not important
Vectors of equal
length & direction are equal
Can translate vectors for convenience (choose ref frame) 2 R q R q 1 R q
x (m) -3 -2 -1 1 2 3 -1 R q -2 -3

5. Adding Vectors: Head-to-Tail
Must have same UNITS (true for scalars also)
Must add magnitudes AND directions..how?
A + B = ?
Head-to-Tail Method B A B
A + B A

6. Adding Vectors: Commutative Property
A + B = B + A ? B A
B + A
A + B A B
YES!
A + B = B + A
Can add vectors in any order

7. Negative of vector = 180o rotation
Subtracting Vectors
A -> -A
Negative of vector = 180o rotation
A - B = A + (-B) B A A -B -B -A
A - B

8. Multiplying & Dividing Vectors by Scalars
2 * A = 2A
-2 * A = -2A A A 2A
-2A x
Ex. v = x/t
t = 3 s v

9. Graphical Vector TechniquesW S E
1 box = 10 km
A plane flies from base camp to lake A a distance 280 km at a direction 20o north of east. After dropping off supplies, the plane flies to lake B, which is 190 km and 30.0o west of north from lake A.
Graphically determine the distance and direction from lake B to the base camp.
190 km
30o
lake B
20o
280 km
lake A
base camp

10. Vector Components Every vector can be described by its components
Component = projection of vector on x- or y-axis
Rx = Rcosq
Ry = Rsinq R Rx Ry x y q
From magnitude (R) and direction (q) of R can determine Rx and Ry y B
= A + B R x A

11. Can determine any vector from its components
Vector Components
Can determine any vector from its components y Ry
R2 = Rx2 + Ry2
R = (Rx2 + Ry2)1/2
tanq = Ry/Rx
= tan-1(Ry/Rx)
-90 < q < 90 R q x Rx

12. Can determine any vector from its components
Vector Components
Can determine any vector from its components
Careful! y
R2 = Rx2 + Ry2
R = (Rx2 + Ry2)1/2
tanq = Ry/Rx
q = tan-1(Ry/Rx)
-90 < q < 90
(-x, +y)
(+x, +y) II I x
III IV
(-x, -y)
(+x, -y)
I, IV: q = tan-1(Ry/Rx)
II, III: q = tan-1(Ry/Rx) + 180o
Important to know direction of vector!

13. Vector Addition: Components
Why are components useful?
When is magnitude of A + B = A + B ? A B
A + B
Rx = Ax + Bx + Cx….
Ry = Ay + By + Cy….
q = tan-1(Ry/Rx)
-90 < q < 90
R = A + B + C…. = ?

14. Vector Addition: Components
lake B
Using components determine the distance and direction from lake B to the base camp.
190 km
30o
Rx = Ax + Bx + Cx….
Ry = Ay + By + Cy….
q = tan-1(Ry/Rx)
-90 < q < 90
lake A
280 km
20o
base camp

15. Vector Components: Problem #2
A man pushing a mop across a floor cause the mop to undergo two displacements. The first has a magnitude of 150 cm and makes an angle of 120o with the positive x-axis. The resultant displacement has a magnitude of 140 cm and is directed at an angle of 35.0o to the positive x-axis. Find the magnitude and direction of the second displacement.

16. Vector Components: Problem #3
An airplane starting from airport A flies 300 km east, then 350 km at 30.0o west of north, and then 150 km north to arrive finally at airport B. The next day, another plane flies directly from A to B in a straight line.
In what direction should the pilot travel in this direct flight?
How far will the pilot travel in the flight?

17. Questions of the Day
1) Can a vector A have a component greater than its magnitude A?
YES
b) NO
2) What are the signs of the x- and y-components
of A + B in this figure?
a) (x,y) = (+,+)
b) (+,-)
c) (-,+)
d) (-,-) A B