VectorsVector quantity has direction as well as magnitude
1.On a diagram, draw one of the vectors – call it D1 - to scale.
2. Next draw the second vector, D2, to scale, placing its tail at the tip of the first vector and being sure its direction is correct.
3. The arrow drawn from the tail of the first vector to the tip of the second represents the sum, or resultant, of the two vectors.
)(1221 lawecommutativVVVV
)()()( 312321 laweassociativVVVVVV
No vector is ever negative in the sense of its magnitude: the magnitude of every vector is positive
Resolving the vector into its components
16 m
You want to measure the height of a building. You stand 2m away from a 3m pole and see that it’s “in line” with the top of the building. You measure 16 m from the pole to the building. What is the height of the building?
EXAMPLES
yyy
xxx
VVV
VVV
21
21
Adding Vectors by Components
1. Draw a diagram2. Choose x and y axes. Choose them in a way that make your
work easier. (E.g. choose one axis along the direction of one of the vectors so that the vector will have only one component).
3. Resolve each vector in x and y components4. Calculate each component using sine and cosine. Be
careful with signs: any component that points along the negative x or y axis gets a negative sign.
5. Add the x components together to get the x component of the resultant. Similar for y:
Vx=V1x+V2x+…Vy=V1y+V2y+…
6. If you want to know the magnitude and direction of the resultant vector,
22yx VVV
x
y
V
Vtan
Mail carrier’s displacement
A rural mail carrier leaves the post office and drives 22.0 km in a northerly direction to the next town. She then drives in a direction 60.00 south of east for 47.0 km to another town. What is her displacement from the post office?
x
y
A
B
C
12
Three vectors are shown in the Figure. Their magnitudes are
02
01 56,28
;31,5.26,44
CBA
Determine the sum of the three vectors. Give the resultant in terms of
a) componentsb) magnitude and angle with x axis
What is the sum of the two vectors and ? a
b
x
y
1
2
a
b
An airplane trip involves three legs, with two stopovers.The first leg is due east for 620 km; the second leg is
southeast (450) for 440 km; and the third leg is at 530 south of west for 550 km. What is the plane’s total displacement?
a) Express the vectors , , and in terms of their components.
1F
2F
3F
x
y
1F
2F
3F
2
3
b) Find . 321 FFF
Three horizontal ropes pull on a large stone stuck in the ground, producing the vector forces and Find the magnitude and direction of a fourth force on the stone that will make the vector sum of the four forces zero.
,A
,B
.C
x
y
)100( NA
30o
)40( NC
53o
)80( NB
30o
Given two vectors, and
a) Find the components of the vector bar
b) Find the magnitude of and the tangent of the angle makes with the x axis.
r
r
Quiz
Given two vectors, and
a) find the magnitude of each vector
b) Write an expression for the vector difference using unit vectors BA
