,Vectors in the Plane
, 7.3 Vectors in the Plane
Introduction What you should learn
Represent vectors as directed
Many quantities in geometry and physics, such as area, time, and temperature, can
line segments.
be represented by a single real number. Other quantities, such as force and velocity,
Write the component forms of
involve both magnitude and direction and cannot be completely characterized by a
vectors.
single real number. To represent such a quantity, you can use a directed line segment,
\
Perform basic vector operations
as shown in Figure 7.11. The directed line segment PQ has initial point P and
\
and represent vectors
terminal point Q. Its magnitude, or length, is denoted by PQ and can be found by
graphically.
using the Distance Formula.
Write vectors as linear
Q combinations of unit vectors.
PQ Terminal Find the direction angles of
point vectors.
P
Use vectors to model and solve
Initial
point real-life problems.
Figure 7.11 Figure 7.12 Why you should learn it
Vectors are used to analyze
Two directed line segments that have the same magnitude and direction are
numerous aspects of everyday life.
equivalent. For example, the directed line segments in Figure 7.12 are all equivalent.
Exercise 99 on page 571 shows
The set of all directed line segments that are equivalent to a given directed line segment
\ \
you how vectors can be used to
PQ is a vector v in the plane, written v = PQ . Vectors are denoted by lowercase,
determine the tension in the cables
boldface letters such as u, v, and w.
of two cranes lifting an object.
EXAMPLE 1 Showing That Two Vectors Are Equivalent
Show that u and v in the figure at the right y
(4, 4)
are equivalent. 4 S
Solution v
3
(1, 2) (3, 2)
From the Distance Formula, it follows 2
\ \
that PQ and RS have the same magnitude. R Q
\ 1
u
PQ = √(3 − 0)2 + (2 − 0)2 = √13 (0, 0)
x
\
RS = √(4 − 1) + (4 − 2) = √13
2 2 P 1 2 3 4
Moreover, both line segments have the same direction, because they are both directed
toward the upper right on lines having the same slope.
\ 2−0 2
Slope of PQ = =
3−0 3
\ 4−2 2
Slope of RS = =
4−1 3
\ \
Because PQ and RS have the same magnitude and direction, u and v are equivalent.
Show that u and v in the figure at the right y
are equivalent.
4
(5, 3)
3
(2, 2) v S
2 R
(3, 1)
1 u Q
x
P 1 2 3 4 5
(0, 0)
, 7.3 Vectors in the Plane
Introduction What you should learn
Represent vectors as directed
Many quantities in geometry and physics, such as area, time, and temperature, can
line segments.
be represented by a single real number. Other quantities, such as force and velocity,
Write the component forms of
involve both magnitude and direction and cannot be completely characterized by a
vectors.
single real number. To represent such a quantity, you can use a directed line segment,
\
Perform basic vector operations
as shown in Figure 7.11. The directed line segment PQ has initial point P and
\
and represent vectors
terminal point Q. Its magnitude, or length, is denoted by PQ and can be found by
graphically.
using the Distance Formula.
Write vectors as linear
Q combinations of unit vectors.
PQ Terminal Find the direction angles of
point vectors.
P
Use vectors to model and solve
Initial
point real-life problems.
Figure 7.11 Figure 7.12 Why you should learn it
Vectors are used to analyze
Two directed line segments that have the same magnitude and direction are
numerous aspects of everyday life.
equivalent. For example, the directed line segments in Figure 7.12 are all equivalent.
Exercise 99 on page 571 shows
The set of all directed line segments that are equivalent to a given directed line segment
\ \
you how vectors can be used to
PQ is a vector v in the plane, written v = PQ . Vectors are denoted by lowercase,
determine the tension in the cables
boldface letters such as u, v, and w.
of two cranes lifting an object.
EXAMPLE 1 Showing That Two Vectors Are Equivalent
Show that u and v in the figure at the right y
(4, 4)
are equivalent. 4 S
Solution v
3
(1, 2) (3, 2)
From the Distance Formula, it follows 2
\ \
that PQ and RS have the same magnitude. R Q
\ 1
u
PQ = √(3 − 0)2 + (2 − 0)2 = √13 (0, 0)
x
\
RS = √(4 − 1) + (4 − 2) = √13
2 2 P 1 2 3 4
Moreover, both line segments have the same direction, because they are both directed
toward the upper right on lines having the same slope.
\ 2−0 2
Slope of PQ = =
3−0 3
\ 4−2 2
Slope of RS = =
4−1 3
\ \
Because PQ and RS have the same magnitude and direction, u and v are equivalent.
Show that u and v in the figure at the right y
are equivalent.
4
(5, 3)
3
(2, 2) v S
2 R
(3, 1)
1 u Q
x
P 1 2 3 4 5
(0, 0)
