MA 15400
Lesson 20
Section 8.3
Vectors
We use PQ to denote the vector with initial point P and terminal point Q.
The name of this
vector is v.
v
Every vector has 2
components; a
direction and a
magnitude, which
is represented by
the length from P
to the arrow.
Q
P
A vector is a force that has both magnitude and direction.
The direction is indicated by the arrow at the terminal point.
The magnitude is the length of the segment representing the vector. || v ||
These are not
absolute value
bars.
v (magnitude of the vector v) represents the length of the vector.
A vector that represents a pull or push of some type is a force vector. Examples of force vectors
include a car (magnitude is speed and direction is obvious) and a thrown ball (force would be
determined by size or weight of ball and speed with which it was thrown).
A single force that represents the combined forces of two combined vectors is a resultant force.
An example of a resultant force would be one car hitting another car. Each car has a force.
Where the hit car ends up after being hit by the second car is a result of its own force and the
force of the car that hit it. We use a parallelogram to determine the resultant force. The
diagonal of the parallelogram represents the resultant force.
Vectors a and b combined together make a resultant vector r
represented by the diagonal of the parallelogram formed.
r
a
b
Parallelograms are reviewed on the here and the picture is shown on the next page. Adjacent
angles of a parallelogram are supplementary, opposite angles are congruent. Opposite sides are
parallel and congruent. Diagonals will not be congruent.
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MA 15400
Lesson 20
Section 8.3
Vectors
Vectors can be doubled, tripled, halved, etc. This new vector could be represented by
multiplying the original vector by a number m. If a vector is multiplied by 2, the result is a
vector in the same direction, but twice the length. If a vector is multiplied by -2, the result is a
vector in the opposite direction and twice the length.
mv is a scalar multiple of the vector v. If m > 0 then it has the same direction as v.
If m < 0 then it has the opposite direction as v.
By placing a vector’s initial point at the origin, the xy-plane is used to represent the
vector. The numbers a1 and a2 are the components of vector < a1, a2 >.
The direction of the vector is determined by graphing a1 and a2.
The magnitude of the vector is the length of the segment.
Vector d shown would
have the components
10, 3 . Components
is comparable to an
ordered pair.
d
(10, 3)
Magnitude of a vector:
a  a1 , a 2  a 12  a 22
Addition of vectors:
a1 , a2  b1 , b2  a1  b1 , a2  b2
Subtraction of vectors:
a1 , a2  b1 , b2  a1  b1 , a2  b2
Scalar multiple of a vector: m a1 , a2  ma1 , ma2
2
The magnitude of vector
d could be found by
using the Pythagorean
Thm.
MA 15400
Lesson 20
Section 8.3
Vectors
Ex 1) Find the magnitude of the vector 2,5
Ex 2) Find the addition vector and subtraction vectors below.
1,9  4, 2
6, 8  5, 3
When adding vectors the
result would be the
diagonal of the
parallelogram resulting
from the two given
vectors.
Ex 3) Find the components of the following vectors: 4 3, 7 and  3 2,1
The first would be
a vector 4 times as
long in the same
direction. The
second, a vector 3
times as long in
the opposite
direction.
A unit vector is any vector with a length 1 unit. However, there are two special unit vectors that
are 1 unit long from the origin on the positive x-axis direction and on the positive y-axis
direction.
The i here is not
Special vectors i = 1,0 , j = 0,1 are unit vectors of magnitude 1.
the imaginary
unit.
Using the special unit vectors above leads to a second way to denote vectors.
An alternate way of denoting vectors: a  a1 , a2  a1i  a2 j
Ex 4) Write each vector in the alternate way.
4,5 
0, 6 
4i  2 j 
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MA 15400
Lesson 20
Section 8.3
Vectors
Ex 5) Sketch vectors a and b, then find and sketch 2a, – b, a + b, a – b, and 3a + 2b
a  2i  3 j , b  4i  2 j
a   3,4 , b  4,2
y
y
x
x
a+b=
a+b=
a–b=
a–b=
3a + 2b =
3a + 2b =
2a – b =
2a – b =
Note: The vector a + b is the diagonal (resultant force) of a parallelogram formed by vectors a
and b.
Find the magnitude of a and the smallest positive angle  from the positive x-axis to the vector
OP that corresponds to a.
y
a  2 3,2
tan  

2
1

2 3
3
R 
x
4
y
x

6
  

6

7
6
MA 15400
y
Lesson 20
Section 8.3
Vectors
a  2i  3j
x
The vectors a and b represent two forces acting at the same point, and  is the smallest positive
angle between a and b. Approximate the magnitude of the resultant force.
a = 40 lb, b = 70 lb  = 45
We will use the
law of cosines.
However, we
need to complete
the
parallelogram to
find the angle.
a = 30 kg, b = 50 kg,  = 150
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