8-7 Vectors
You used trigonometry to find side lengths and
angle measures of right triangles.
• Perform vector operations geometrically.
• Perform vector operations on the coordinate
plane.
Definition
B
Terminal point or tip
A
Initial point or tail
A vector can be represented as a “directed” line
segment, useful in describing paths.
A vector looks like a ray, but it is NOT!!
A vector has both direction and magnitude (length).
Direction and Length
From the school
entrance, I went three
blocks north.
The distance
(magnitude) is:
Three blocks
The direction is:
North
Direction and Magnitude
The magnitude of AB is the distance
between A and B.
The direction of a vector is measured
counterclockwise from the horizonal (positive
x-axis).
B
N
B
60°
A
45°
A
E
W
S
Are vectors really used?
http://www.nbclearn.com/portal/site/learn/sc
ience-of-nfl-football
Drawing Vectors
Draw vector YZ with direction of 45° and
length of 10 cm.
1.Draw a horizontal dotted line
45°
2.Use a protractor to draw 45°
3.Use a ruler to draw 10 cm
4.Label the points
Y
Z
A. Use a ruler and a protractor to draw each vector.
Include a scale on each diagram.
= 80 meters at 24° west of north
Using a scale of 1 cm : 50 m, draw and label an
80 ÷ 50 or 1.6-centimeter arrow 24º west of the northsouth line on the north side.
Answer:
B. Use a ruler and a protractor to draw each vector.
Include a scale on each diagram.
= 16 yards per second at 165° to the horizontal
Using a scale of 1 cm : 8 yd/s, draw and label a 16 ÷ 8
or 2-centimeter arrow at a 165º angle to the horizontal.
Answer:
Using a ruler and a protractor, draw a vector to
represent
feet per second 25 east of north.
Include a scale on your diagram.
A.
C.
B.
D.
Resultant = Vector Sum
A path or trip that consists of several segments
can be modeled by a sequence of vectors.
The endpoint of one vector is the origin of the
next vector in the chain. The figure shows a
ship’s path from point M to point N that
consists of five vectors.
U
T
S
V
M
N
Resultant (Vector Sum)
What is the shortest path from M to N?
Write the vector sum for the boat’s trip starting
with MS
U
T
S
V
M
N
For resultants (vector sums), the
following is true:
XY + YZ = XZ
Y
Z
X
p. 601
Types of Vectors
• Parallel vectors have
the same or opposite
direction but not
necessarily the same
magnitude (length)
• Opposite vectors
have the same
magnitude but opposite
direction.
• Equivalent vectors
have the same
magnitude and
direction.
Find the Resultant of Two Vectors
Copy the vectors. Then find
b
a
Subtracting a vector is equivalent to
adding its opposite.
Method 1 Use the parallelogram method.
Step 1
, and translate it so
that its tail touches the tail of .
a
–b
a
–b
Step 2
Complete the parallelogram.
Then draw the diagonal.
a –b
a
–b
Method 2 Use the triangle method.
Step 1
, and translate it so
that its tail touches the tail of .
–b
Step 2
Draw the resultant
vector from the tail of
to the tip of – .
a
–b
a
Answer:
a – b
a – b
Copy the vectors. Then find
A.
B.
a–b
C.
a–b
D.
a–b
a–b
a
b
Vectors on the Coordinate Plane
Write the component form of
.
Find the change of x-values and the
corresponding change in y-values.
Component form of vector
Simplify.
Write the component form of
A.
B.
C.
D.
.
Assignment
Worksheet 3-1C
8-7 Vectors day 2
You used trigonometry to find side lengths and
angle measures of right triangles.
• Perform vector operations geometrically.
• Perform vector operations on the coordinate
plane.
Find the Magnitude and Direction of a Vector
Find the magnitude and direction of
Step 1
Use the Distance Formula to find the
vector’s magnitude.
Distance Formula
(x1, y1) = (0, 0) and
(x2, y2) = (7, –5)
Simplify.
Use a calculator.
Step 2
Use trigonometry to find
the vector’s direction.
Definition of inverse tangent
Use a calculator.
Answer:
Find the magnitude and direction of
A. 4; 45°
B. 5.7; 45°
C. 5.7; 225°
D. 8; 135°
p. 603
Scalar – a constant multiplied by a vector
Scalar multiplication – multiplication of a vector by a
scalar (dilation)
Find each of the following for
and
. Check your answers graphically.
A.
Solve Algebraically
Check Graphically
Find each of the following for
and
. Check your answers graphically.
B.
Solve Algebraically
Check Graphically
Find each of the following for
and
. Check your answers graphically.
C.
Solve Algebraically
Check Graphically
A.
B.
C.
D.
CANOEING Suppose a person is canoeing due east
across a river at 4 miles per hour. If the river is flowing
south at 3 miles per hour, what is the resultant speed
and direction of the canoe?
Draw a diagram. Let
represent the resultant vector.
The component form of the vector representing the velocity of
the canoe is 4, 0, and the component form of the vector
representing the velocity of the river is 0, –3. The resultant
vector is 4, 0 + 0, –3 or 4, –3, which represents the
resultant velocity of the canoe. Its magnitude represents the
resultant speed.
Use the Distance Formula to find the resultant speed.
Distance Formula
(x1, y1) = (0, 0) and
(x2, y2) = (4, –3)
The resultant speed of the canoe is 5 miles per hour.
Use trigonometry to find the resultant direction.
Definition of inverse tangent
Use a calculator.
The resultant direction of the canoe is about 36.9° south
of due east.
Answer: Therefore, the resultant speed of the canoe
is 5 mile per hour at an angle of about
90° – 36.9° or 53.1° east of south.
8-7 Assignment