Download 7.4 Vectors, Operations, and the Dot Product

7.4 Vectors, Operations, and the Dot Product
Quantities such as area, volume, length, temperature, and speed have magnitude only and can be
completely characterized by a single real number with a unit of measure such as square feet,
cubic feet, miles, degrees, or miles per hour. A quantity of this type is a scalar quantity, and the
corresponding real number is a scalar. Concepts such as velocity, acceleration, force,
momentum, and electric field have both magnitude and direction and are often represented by a
directed line segment. Another name for a directed line segment is a vector.
To find the sum of two vectors
point of vector
. The sum
and , we place the initial point of vector
+ , is the vector with the same initial point as
at the terminal
and the same
terminal point as .
1. Sketch the graph of
and
+
given the vectors
and
shown below.
Solution:
+
Another method to sketch the graph of the sum of two vectors is to use the parallelogram rule.
Place vectors
has
and
is the sum
and
so that their initial points coincide. Then, complete a parallelogram that
as two sides. The diagonal of the parallelogram with the same initial point as
+
.
2. Sketch the graph of 2
given the vectors
and
shown below.
2
2
and
Algebraic Interpretation of Vectors. A vector with its initial point at the origin is called a
position vector. A position vector with its endpoint at the point , is written as
, .
The numbers a and b are the horizontal and vertical components of the vector . The positive
angle between the x-axis and a position vector is called vector’s direction angle.
(a,b)
,
Magnitude and Direction Angle of a Vector
,
The magnitude (length) of vector
, is given by
| |
satisfies tan
The direction angle
In examples 3 and 4, find the magnitude and direction angle for each vector.
3. 15, 8
8
Solution: The magnitude is 15
The direction angle satisfies tan
so
√289
17
tan
vector is in the 4th quadrant, the direction angle is 360°
4.
28.07°. Since the position
28.07°
331.93°.
7,24
Solution: The magnitude is
The direction angle satisfies tan
7
24
so
25
tan
vector is in the 2 quadrant, the direction angle is 180°
nd
73.74°. Since the position
73.74°
Horizontal and Vertical Components
The horizontal and vertical components, respectively, of a vector
direction angle are given by
| | cos and
| | sin
| | cos , | | sin
,
That is
106.26°.
having magnitude | | and
5. Write the vector in the form
,
.
220°
8
Solution:
,
| | cos , | | sin
8 cos 220° , 8 sin 220°
6.13, 5.14
Properties of Parallelograms
 A parallelogram is a quadrilateral whose opposite sides are parallel
 The opposite sides and opposite angles of a parallelogram are equal, and
adjacent angles of a parallelogram are supplementary.
 The diagonals of a parallelogram bisect each other, but do not necessarily
bisect the angles of the parallelogram.
6. Two forces of magnitude 116 lb and 139 lb act a point in the plane. The angle between the two
forces is 140050’. Find the magnitude of the resultant vector.
Solution: Use the parallelogram rule for adding vectors.
139
116
Adjacent angles of a parallelogram are supplementary, therefore the angle between the two sides
of the parallelogram is 180° 140°50
39°10
By the law of cosines
2 cos
116
139
2 116 139 cos 39°10
116
88.17
139
2 116 139 cos 39°10
7. Use the parallelogram rule to find the magnitude of the resultant force for the two forces
shown in the figure.
85 lb
500
190 lb
Solution : The angle between the two sides of the parallelogram is the supplement of the angle
between the vectors, that is 1300.
By the law of cosines:
2 cos
85
190
2 85 190 cos 130°
85
190
253.15
2 85 190 cos 130°
The Unit Basis Vectors
The unit basis vectors
are defined as:
1,0
0,1
Any position vector can be written as a linear combination of the unit basis vectors.
,
,0
0,
1,0
0,1
In examples 8 and 9, write each position vector in the form
8. 7, 5
Solution: 7
5
9.
4,0
Solution: 4
Dot Product
The dot product of the two vectors
· and defined as
·
,
,
and
is denoted by
In problems 10 -12, find the dot product for each pair of vectors.
10. 6, 2 , 3,5
Solution: 6, 2 · 3,5
6·3
2 ·5
18 10
8
11. 3,8 , 3, 5
Solution: 3,8 · 3, 5
3·3 8 5
9
40
49
12. 5
12 , 3
2
Solution: 5
12 · 3
2
5 3
12 2
15
Geometric Interpretation of the Dot Product
If is the angle between two nonzero vectors
·
cos
| || |
13. Find the angle between 5,2
Solution:
·
5 6
2 3
cos
| || | √5
2
6
=
24
cos
cos
√
, then
6,3
3
131.6°
√1305
·
9
√
√1305
24
Theorem: If
24
0 then the vectors
14. Determine if 1,2
Solution: 1 6
2 3
are orthogonal (perpendicular).
6,3 are orthogonal.
6 6 0, thus the vectors are orthogonal.