4. Draw the Displacement vector for the
following
A.
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b)
EN
B
Answers
Answer:
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Answer: Displacement and velocity in two or three dimensions are straightforward extensions of the one-dimensional definitions. However, now they are vector quantities, so calculations with them have to follow the rules of vector algebra, not scalar algebra.
1
located at
P
1
with position vector
→
r
(
t
1
)
.
At a later time
t
2
,
the particle is located at
P
2
with position vector
→
r
(
t
2
)
. The displacement vector
Δ
→
r
is found by subtracting
→
r
(
t
1
)
from
→
r
(
t
2
)
:
Δ
→
r
=
→
r
(
t
2
)
−
→
r
(
t
1
)
.
Vector addition is discussed in Vectors. Note that this is the same operation we did in one dimension, but now the vectors are in three-dimensional space.
An x y z coordinate system is shown, with positive x out of the page, positive y to the right, and positive z up. Two points, P 1 and P 2 are shown. The vector r of t 1 from the origin to P 1 and the vector r of t 2 from the origin to P 2 are shown as purple arrows. The vector delta r is shown as a purple arrow whose tail is at P 1 and head at P 2.
Figure 4.3 The displacement
Δ
→
r
=
→
r
(
t
2
)
−
→
r
(
t
1
)
is the vector from
P
1
to
P
2
.Displacement and velocity in two or three dimensions are straightforward extensions of the one-dimensional definitions. However, now they are vector quantities, so calculations with them have to follow the rules of vector algebra, not scalar algebra.
Displacement Vector
To describe motion in two and three dimensions, we must first establish a coordinate system and a convention for the axes. We generally use the coordinates x, y, and z to locate a particle at point P(x, y, z) in three dimensions. If the particle is moving, the variables x, y, and z are functions of time (t):
x
=
x
(
t
)
y
=
y
(
t
)
z
=
z
(
t
)
.
The position vector from the origin of the coordinate system to point P is
→
r
(
t
)
.
In unit vector notation, introduced in Coordinate Systems and Components of a Vector,
→
r
(
t
)
is
→
r
(
t
)
=
x
(
t
)
^
i
+
y
(
t
)
^
j
+
z
(
t
)
^
k
.
(Figure) shows the coordinate system and the vector to point P, where a particle could be located at a particular time t. Note the orientation of the x, y, and z axes. This orientation is called a right-handed coordinate system (Coordinate Systems and Components of a Vector) and it is used throughout the chapter.
An x y z coordinate system is shown, with positive x out of the page, positive y to the right, and positive z up. A point P, with coordinates x of t, y of t, and z of t is shown. All of P’s coordinates are positive. The vector r of t from the origin to P is also shown as a purple arrow. The coordinates x of t, y of t and z of t are shown as dashed lines. X of t is a segment in the x y plane, parallel to the x axis, y of t is a segment in the x y plane, parallel to the y axis, and z of t is a segment parallel to the z axis.
Figure 4.2 A three-dimensional coordinate system with a particle at position P(x(t), y(t), z(t)).
With our definition of the position of a particle in three-dimensional space, we can formulate the three-dimensional displacement. (Figure) shows a particle at time
t
1
located at
P
1
with position vector
→
r
(
t
1
)
.
At a later time
t
2
,
the particle is located at
P
2
with position vector
→
r
(
t
2
)
. The displacement vector
Δ
→
r
is found by subtracting
→
r
(
t
1
)
from
→
r
(
t
2
)
:
Δ
→
r
=
→
r
(
t
2
)
−
→
r
(
t
1
)
.
Vector addition is discussed in Vectors. Note that this is the same operation we did in one dimension, but now the vectors are in three-dimensional space.
An x y z coordinate system is shown, with positive x out of the page, positive y to the right, and positive z up. Two points, P 1 and P 2 are shown. The vector r of t 1 from the origin to P 1 and the vector r of t 2 from the origin to P 2 are shown as purple arrows. The vector delta r is shown as a purple arrow whose tail is at P 1 and head at P 2.
Figure 4.3 The displacement
Δ
→
r
=
→
r
(
t
2
)
−
→
r
(
t
1
)
is the vector from
P
1
to
P
2
.
Explanation: