### Addition and Subtraction of Vectors

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Lecture #3 - PHYC 160
Multiplying a vector by a scalar
•  If c is a scalar, the
→
product cA has
magnitude |c|A.
•  Figure 1.15 illustrates
multiplication of a vector
by a positive scalar and a
negative scalar.
In 3-dimensions
A
A
Axiˆ
Ay ˆj
Ax
2
Az kˆ
Ay
2
Az
2
Unit vectors—Figures 1.23–1.24
•  A unit vector has a magnitude
of 1 with no units.
•  The unit vector î points in the
\$
+x-direction, \$jj points in the +y\$k points in the
direction, and \$
k
+z-direction.
•  Any vector can be expressed
in terms of its components as
→
\$k.
A =Axî+ Ay \$\$jj + Az \$
k
Q1.7
The angle θ is measured
counterclockwise from the
positive x-axis as shown.
For which of these vectors
is θ greatest?
A. 24 iˆ +18 ˆj
B. −24 iˆ − 18 ˆj
ˆ
ˆ
C. −18 i + 24 j
D. −18 iˆ − 24 ˆj
Adding more than two vectors graphically—Figure 1.13
•  The vectors can be added in any order.
Subtracting vectors
•  Figure 1.14 shows how to subtract vectors.
Addition of two vectors at right angles
•  First add the vectors graphically.
•  Then use trigonometry to find the magnitude and direction of the
sum.
You didnt know
this,
but you are
when you give
directions on a
map
You
CPS Question 3-1
• Which vector most closely represents A B with A
and
?
B
A.
B.
C.
D.
Components of a vector—Figure 1.17
•  Adding vectors graphically provides limited accuracy. Vector
components provide a general method for adding vectors.
•  Any vector can be represented by an x-component Ax and a ycomponent Ay.
•  Use trigonometry to find the components of a vector: Ax = Acos θ and
Ay = Asin θ, where θ is measured from the +x-axis toward the +y-axis.
Calculations using components
•  We can use the components of a vector to find its magnitude
A
A = Ax2 + Ay2 and tanθ = y
and direction:
A
x
•  We can use the components of a
set of vectors to find the components
of their sum:
Rx = Ax + Bx + Cx +L , Ry = Ay + By +Cy +L
•  Refer to Problem-Solving
Strategy 1.3.
Q1.2
Consider the
vectors shown.
Which is a correct
statement
! !
A+ B?
A. x-component > 0, y-component > 0
B. x-component > 0, y-component < 0
C. x-component < 0, y-component > 0
D. x-component < 0, y-component < 0
E. x-component = 0, y-component > 0
Q1.3
Consider the
vectors shown.
Which is a correct
statement
! !
A − B?
A. x-component > 0, y-component > 0
B. x-component > 0, y-component < 0
C. x-component < 0, y-component > 0
D. x-component < 0, y-component < 0
E. x-component = 0, y-component > 0
Q1.5
Which
!
!of the following statements is correct for any two vectors
A and B ?
! !
A. The magnitude of A − B
! !
B. The magnitude of A − B
! !
C. The magnitude of A − B
! !
D. The magnitude of A − B
! !
E. The magnitude of A − B
is A – B.
is A + B.
is greater than or equal to |A – B|.
! !
is less than the magnitude of A + B.
2
2
A
+
B
is
.
The scalar product—Figures 1.25–1.26
•  The scalar product
(also called the dot
product ) of two
vectors
r r is
AgB = AB cosφ.
•  Figures 1.25 and
1.26 illustrate the
scalar product.
Q1.10
Consider the two vectors
!
A = 3iˆ + 4 ˆj
!
B = −8iˆ + 6 ˆj
! !
What is the dot product A • B ?
A. zero
B. 14
C. 48
D. 50
E. none of these