Five-Minute Check (over Lesson 8-2)
Then/Now
New Vocabulary
Key Concept: Dot Product of Vectors in a Plane
Key Concept: Orthogonal Vectors
Example 1: Find the Dot Product to Determine Orthogonal Vectors
Key Concept and Proof: Properties of the Dot Product
Example 2: Use the Dot Product to Find Magnitude
Key Concept and Proof: Angle Between Two Vectors
Example 3: Find the Angle Between Two Vectors
Key Concept and Proof: Projection of u onto v
Example 4: Find the Projection of u onto v
Example 5: Projection with Direction Opposite v
Example 6: Real-World Example: Use a Vector Projection to Find a Force
Example 7: Real-World Example: Calculate Work
Over Lesson 8-2
Find the component form and magnitude of
initial point A(−3, 7) and terminal point B(6, 2).
A.
B.
C.
with
Over Lesson 8-2
Find the component form and magnitude of
with initial point A(2, 5) and terminal point B(8, –3).
A. 6, –8; 10
B. –6, 8; 10
C.
D.
Over Lesson 8-2
Find 2f − 3g + 4h if f = −7, −2, g = 3, 1, and
h = 9, −1.
A. 13, –11
B. –11, 13
C. 41, –11
D. 41, 0
Over Lesson 8-2
Find a unit vector with the same direction as
v = −1, 5.
A.
B.
C.
Over Lesson 8-2
Which of the following represents the direction
angle of the vector 2i − 8j?
A. 75.96°
B. 104.04°
C. 284.04°
D. 345.96°
You found the magnitudes of and operated with
algebraic vectors. (Lesson 8-2)
• Find the dot product of two vectors and use the dot
product to find the angle between them.
• Find the projection of one vector onto another.
• dot product
• orthogonal
• vector projection
• work
Find the Dot Product to Determine Orthogonal
Vectors
A. Find the dot product of u and v if u = –3, 4 and
v = 3, 6. Then determine if u and v are
orthogonal.
u ● v = –3(3) + 4(6)
= 15
Since u ● v ≠ 0, u and v are not
orthogonal, as illustrated.
Answer: 15; not orthogonal
Find the Dot Product to Determine Orthogonal
Vectors
B. Find the dot product of u and v if u = 2, 7 and
v = –14, 4 . Then determine if u and v are
orthogonal.
u ● v = 2(–14) + 7(4)
=0
Since u ● v = 0, u and v are
orthogonal, as illustrated.
Answer: 0; orthogonal
Find the dot product of u = 4, –1 and v = –3, –5.
Then determine if u and v are orthogonal.
A. –7; yes
B. –7; no
C. 17; yes
D. 17; no
Use the Dot Product to Find Magnitude
Use the dot product to find the magnitude of
a = –6, 5.
Since |a|2 = a ● a, then |a| =
.
a = –6, 5
Simplify.
The magnitude of a is
Answer:
or about 7.81.
or about 7.81
Use the dot product to find the magnitude of
v = –4, –1.
A.
B. 17
C.
D. 15
Find the Angle Between Two Vectors
A. Find the angle θ between u = –3, –5 and
v = 2, –3 to the nearest tenth of a degree.
Angle between two
vectors
u = –3, –5 and
v = 2, –3
Evaluate.
Simplify.
Find the Angle Between Two Vectors
Solve for θ.
The measure of the angle between u and v
is about 64.7°.
Answer: 64.7°
Find the Angle Between Two Vectors
B. Find the angle θ between u = 1, –4 and
v = 2, 6 to the nearest tenth of a degree.
Angle between two
vectors
u = 1, –4 and
v = 2, 6
Evaluate.
Simplify.
Find the Angle Between Two Vectors
Solve for .
The measure of the angle between u and v
is about 147.5°.
Answer: 147.5°
Find the angle between vectors v = –3, –1 and
w = –5, 2 to the nearest tenth of a degree.
A. 139.8°
B. 69.9°
C. 40.2°
D. 3.4°
Find the Projection of u onto v.
Find the projection of u = –1, 5 onto v = 4, 6.
Then write u as the sum of two orthogonal vectors,
one of which is the projection of u onto v.
Step 1 Find the projection of u onto v.
Projection of u onto v
u = –1, 5 onto
v = 4, 6
Evaluate.
Find the Projection of u onto v.
Scalar multiplication
Step 2 Find w2.
Since u = w1 + w2, w2 = u – w1.
w2 = u – w1
= u – projvu
= –1, 5 – 2, 3
= –3, 2
Therefore, projvu is w1 = 2, 3 as shown on the next
slide, and u = 2, 3 + –3, 2 .
Find the Projection of u onto v.
Answer: projvu = 2, 3; u = 2, 3 + –3, 2
Find the projection of u = 4, –3 onto v = 1, 1.
Then write u as a sum of two orthogonal vectors,
one of which is the projection of u onto v.
A.
projvu =
B.
projvu =
C.
projvu =
D.
projvu =
Projection with Direction Opposite v
Find the projection of u = 4, 2 onto v = –3, 5.
Then write u as the sum of two orthogonal vectors,
one of which is the projection of u onto v.
Notice that the angle between u and v is obtuse, so
the projection of u onto v lies on the vector opposite v
or –v, as shown above.
Projection with Direction Opposite v
Step 1 Find the projection of u onto v.
Projection of u onto v
u = 4, 2 onto
v = –3, 5
Evaluate.
Scalar multiplication
Projection with Direction Opposite v
Step 2 Find w2.
Since u = w 1 + w2, w2 = u – w 1.
w2 = u – w 1
= u – projvu
Projection with Direction Opposite v
Therefore, projvu is w1
and u
Answer:
as shown,
.
Find the projection of u = 3, –2 onto v = 4, 3.
Then write u as the sum of two orthogonal vectors,
one of which is the projection of u onto v.
A. projvu =
B. projvu =
C. projvu =
D. projvu =
Use a Vector Projection to Find
a Force
BOULDERS A 10,000-pound boulder sits on a
mountain at an incline of 60°. Ignoring the force of
friction, what force is required to keep the boulder
from rolling down the mountain?
The weight of the boulder is the force exerted due to
gravity, F = 0, –10,000. To find the force –w1
required to keep the boulder from rolling down the
mountain, project F onto a unit vector v in the
direction of the side of the mountain.
Use a Vector Projection to Find
a Force
Step 1 Find a unit vector v in the direction of the
mountain.
v = |v| (cos θ), |v| (sin θ)
Component form of v
in terms of |v| and θ
Use a Vector Projection to Find
a Force
= 1(cos 60°), 1(sin 60°) or
|v| = 1 and
θ = 60°
Step 2 Find w1, the projection of F onto unit vector v,
projvF.
Projection of F
onto v
Since v is a unit
vector, |v| = 1.
Simplify.
Use a Vector Projection to Find
a Force
F = 0, –10,000
and v =
Find the dot
product.
The force required is –w1 = –(8660.3v) or 8660.3v.
Since v is a unit vector, this means that this force has
a magnitude of about 8660.3 pounds and is in the
direction of the side of the mountain.
Answer: about 8660.3 lb
TRUCKS A 5000-pound truck sits on a hill inclined
at a 15° angle. Ignoring the force of friction, what
force is required to keep the truck from rolling
down the hill?
A. 1294.1 lb
B. 2588.2 lb
C. 4829.6 lb
D. 5000 lb
Calculate Work
MOWING A person pushes a reel mower with a
constant force of 40 newtons at a constant angle
of 45°. Find the work done in joules moving the
mower 12 meters.
Calculate Work
Method 1 Use the projection formula for work.
The magnitude of the projection of F onto
|F| cos θ = 40 cos 45°. The magnitude of
directed distance, is 12.
W=
is
, the
Projection formula for work
= (40 cos 45°)(12)
or about 339.4
Substitution
Calculate Work
Method 2 Use the dot product formula for work.
The component form of the force vector F in terms of
magnitude and direction angle given is 40 cos (–45°),
40 sin (–45°). The component form of the directed
distance the mower is moved is 12, 0.
W =F●
Dot product
formula for work
= 40 cos (–45°), 40 sin (–45°) ● 12, 0 Substitution
= [40 cos (–45°)](12) or about 339.4
Dot product
Therefore, the person does about 339.4 joules of work
pushing the mower.
Answer: about 339.4 joules
CRATE A person pushes a crate along the floor
with a constant force of 30 newtons at a constant
angle of 30°. Find the work done in joules moving
the crate 8 meters.
A. 26.0 joules
B. 120.0 joules
C. 207.8 joules
D. 678.8 joules