Homework Solutions
Geometry CP, Mar 8
Vectors
Book Section: 8-7
Essential Question: What are vectors, what are their components, and
how can I find and use them?
Standards: CCSS G.SRT.6, .7, .8, G.MG.3; G-3.10, .12
Definitions
• Scalar quantity – a quantity described by a real number,
which quantifies a magnitude or size.
• Vector quantity – A quantity that has both a magnitude
and a direction.
• Magnitude of a vector – The length of a vector from its
initial to terminal point.
• Direction of a vector – a direction expressed as an angle
it forms with the horizontal or as a measurement between
0-90 degrees east or west of the north-south line.
• Resultant – The sum or difference of two vectors.
Vector Addition - Methods
Example 1
Find the Resultant of Two Vectors
Copy the vectors. Then find
a
b
Subtracting a vector is equivalent to adding its opposite.
Example 1
Find the Resultant of Two Vectors
Method 1
Use the parallelogram method.
Step 1
a
–b
a
–b
Example 1
Find the Resultant of Two Vectors
Step 2 Complete the parallelogram. Then draw
the diagonal.
a –b
a
–b
Example 1
Find the Resultant of Two Vectors
Method 2
Use the triangle method.
–b
a
Example 1
Find the Resultant of Two Vectors
–b
a
Answer:
a – b
a – b
Example 2
Copy the vectors. Then find
a
A.
B.
a–b
C.
a–b
D.
a–b
a–b
b
Vector Component Form
• A vector is in standard position if its initial point is at
the origin. A vector can be uniquely be described by its
terminal point, P(x, y).
• To describe a vector with any initial point, you can use
the component form  x, y , which describes in terms of
its horizontal component x and vertical component y.
• To write the component form of a vector with initial
point (x1, y1) and terminal point (x2, y2), find
 x2  x1 , y2  y1 
Examples
Try PQ
Example 3
Write a Vector in Component Form
Write the component form of
.
Example 3
Write a Vector in Component Form
Find the change of x-values and the corresponding
change in y-values.
Component form of vector
Simplify.
Example 4
Write the component form of
A.
B.
C.
D.
.
Magnitude and Direction
• The magnitude of a vector, abbreviated as | r | of vector
r, can be found using the distance formula.
• The direction can be found using trigonometric ratios.
Example
Example 5
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.
Example 5
Step 2
Find the Magnitude and Direction of a Vector
Use trigonometry to find the vector’s
direction.
Graph , its horizontal component, and its vertical component.
Then use the inverse tangent function to
find θ.
Component Form Operations
Example 6
Operations with Vectors
Find each of the following for
and
. Check your answers graphically.
A.
Solve Algebraically
Check Graphically
Example 6
Operations with Vectors
Find each of the following for
and
.
Check your answers graphically.
B.
Solve Algebraically
Check Graphically
Example 7
A.
B.
C.
D.

Examples
Classwork: Textbook p.606, 16-28 Even
Homework: HW Due 3/12, 1-10