P.o.D. – Given the initial and
terminal points of a vector,
write a linear combination of
the standard unit vectors i and j.
1.) (-3,1), (4,5)
2.) (0,-2), (3,6)
3.) (1,-5), (2,3)
4.) (-6,4), (0,1)
5.) Find the magnitude and
direction angle of the vector
v=-5i+4j.
1.)
2.)
3.)
4.)
7i+4j
3i+8j
1i+8j
6i-3j
5.) Magnitude=√41; 𝜃 = 141.3°
6-4: Vectors and Dot Products
Learning Target: I will be able to
find the dot product of two
vectors; find the angle between
two vectors; use vectors to find
the work done by a force.
Dot (Inner) Product of Vectors:
If 𝑢
⃗ = ⟨𝑎, 𝑏⟩ and 𝑣 = ⟨𝑐, 𝑑⟩, then
their dot product is
𝑢
⃗ ⋅ 𝑣 = 𝑎𝑐 + 𝑏𝑑.
- If their dot product is 0, then
the vectors are perpendicular
(orthogonal).
EX: Find each dot product if 𝑢
⃗ =
⟨2, −5⟩, 𝑣 = ⟨4,1⟩, 𝑎𝑛𝑑 𝑤
⃗⃗ = ⟨10,4⟩. Are
any pairs of vectors
perpendicular?
a.) 𝑢
⃗ ∙𝑣
⟨2, −5⟩ ∙ ⟨4,1⟩ = 2(4) + (−5)(1)
=8−5=3
Not perpendicular.
b.) 𝑢
⃗ ∙𝑤
⃗⃗
⟨2, −5⟩ ∙ ⟨10,4⟩ = 2(10) + (−5)(4)
= 20 − 20 = 0
Yes, perpendicular.
c.) 𝑣 ∙ 𝑤
⃗⃗
⟨4,1⟩ ∙ ⟨10,4⟩ = 4(10) + 1(4) = 40 + 4
= 44
Not perpendicular.
*Write a program to find dot
products and orthogonal
vectors.
EX: Find the inner product of v
and w if 𝑣 = ⟨2, −3, −4⟩ 𝑎𝑛𝑑 𝑤
⃗⃗ =
⟨8,3,2⟩. Are v and w
perpendicular?
𝑣∙𝑤
⃗⃗ = 2(8) + (−3)(3) + (−4)(2)
= 16 − 9 − 8 = −1
Not perpendicular.
Hulk Hogan was a former
heavyweight wrestling
champion and actor. Suppose
that Hogan and a tag team
partner are each pulling
horizontally and at a right angle
to each other on the arms of an
opponent. Hulk exerts a force of
180-lbs due north while his
partner exerts a force of 125-lbs
due east.
a.) Draw and label a diagram.
(show on the whiteboard)
b.) Determine the resultant
force.
𝑅2 = 1802 + 1252 = 48025
𝑅 = √48025 ≈ 219.15
c.) Determine the angle the
resultant force makes with
the east-west axis.
180
tan 𝜃 =
125
180
−1
𝜃 = 𝑡𝑎𝑛 (
) ≈ 55.22°
125
Work is the dot product of force
and distance; 𝑊 = 𝐹 ∙ 𝑑
EX: Andy works for UPS.
Suppose he is pushing a cart full
of boxes weighing 95-lbs up a
ramp 10-ft long at an incline of
15 degrees. Find the work done
by gravity as the cart moves the
length of the ramp.
(draw a picture on the
whiteboard)
We first need to find the
component vectors that yields
10-ft.
𝑥
cos 15° =
10
𝑥 = 10 cos 15° ≈ 9.66
𝑦
sin 15° =
10
𝑦 = 10 sin 15° ≈ 2.59
Thus, the component vector is
𝑑 = 9.66𝑖 + 2.59𝑗
Since the weight of the box is 95lbs, its force is represented by
𝐹 = 0𝑖 − 95𝑗
𝑊 =𝐹∙𝑑
= ⟨9.66𝑖 + 2.59𝑗⟩ ∙ ⟨0𝑖 − 95𝑗⟩
= 0 − 246.05
= −246.05 𝑓𝑡 − 𝑙𝑏𝑠.
EX: Justin works for FedEx.
Suppose that he is pushing a
cart full of packages weighing
125-lbs up a ramp 10-ft long at
an incline of 20 degrees. Find the
work done by gravity as the cart
moves the length of the ramp.
(Draw a picture on the
whiteboard).
𝑥
cos 20° =
10
𝑥 = 10 cos 20° ≈ 9.40
𝑦
sin 20° =
10
𝑦 = 10 sin 20° ≈ 3.42
𝑑 = 9.4𝑖 + 3.42𝑗; 𝐹 = 0𝑖 − 125𝑗
𝑊 = ⟨0, −125⟩ ∙ ⟨9.4,3.42⟩
= −427.5 𝑓𝑡 − 𝑙𝑏𝑠
EX: Dr. Rizzo is hanging a sign
for his medical practice. The
sign is held by two support bars
as shown in the figure. If the
bars make a 60 degree angle
with each other and the sign
weighs 100-lbs, what are the
magnitudes of the forces exerted
by the sign on each support bar?
(Draw a picture on the
whiteboard)
100
tan 60° =
⃗⃗⃗
𝐹1
⃗⃗⃗
𝐹1 tan 60° = 100
100
⃗⃗⃗
𝐹1 =
≈ 57.74
tan 60°
100
sin 60° =
⃗⃗⃗
𝐹2
⃗⃗⃗
𝐹2 sin 60° = 100
100
⃗⃗⃗
𝐹2 =
≈ 115.47
sin 60°
Angle Between Two Vectors:
𝑢∙𝑣
cos 𝜃 =
‖𝑢‖‖𝑣 ‖
EX: Find the angle between 𝑢 =
⟨3,0⟩ 𝑎𝑛𝑑 𝑣 = ⟨1,6⟩.
Begin by finding the dot
product.
𝑢 ∙ 𝑣 = 3(1) + 0(6) = 3 + 0 = 3
Now find the magnitude of each
vector.
‖𝑢‖ = √9 + 0 = 3
‖𝑣 ‖ = √1 + 36 = √37
Apply for the formula for the
angle between two vectors.
cos 𝜃 =
3
3√37
3√37 √37
cos 𝜃 =
=
111
37
37
𝜃 = 𝑐𝑜𝑠
≈ 80.54°
37
*Let’s add a part to our program
to find the angle between
vectors.
−1 √
Let’s do some review questions
before tomorrow’s quiz.
a.) A triangular parcel of land
has borders of lengths 60m,
70m, and 82m. Find the
area of the parcel of land.
1
𝑠 = (60 + 70 + 82) = 106
2
𝑠
= √106(106 − 60)(106 − 70)(106 − 82)
= √4212864 ≈ 2052.53 𝑠𝑞. 𝑚𝑒𝑡𝑒𝑟𝑠
b.) An airplane flies 370 miles
from point A to point B
with a bearing of 24
degrees. It then flies 240
miles from point B to point
C with a bearing of 37
degrees. Find the distance
and bearing from point A
to point C.
(show a picture and work on
the whiteboard)
606.3 miles; 29.1 degrees.
c.) Find the angle between the
vectors u=-1i+5j and
v=3i-2j.
(show work on the whiteboard)
135 degrees.
d.) The lengths of the
diagonals of a
parallelogram are 30m and
40m. Find the lengths of
the sides of the
parallelogram if the
diagonals intersect at an
angle of 34 degrees.
(show work on the whiteboard)
11.3m, 33.5m
Some More Stuff on Vectors:
EX: A recently built world class
cruise ship stretches 964.5 ft in
length. Suppose that the ship
leaves port and sails for 80-mi in
a direction 50 degrees north of
due east. Draw a picture and
find the magnitude of the
horizontal and vertical
components.
(Draw a picture on the
whiteboard)
ℎ
cos 50° =
80
ℎ = 80 cos 50° ≈ 51.42𝑚𝑖
𝑣
sin 50° =
80
𝑣 = 80 sin 50° ≈ 61.28
EX: Suppose there is a building
1508-ft tall. Suppose that a
piling for this building is being
pushed by two bulldozers at
exactly the same time. One
bulldozer exerts a force of 900lbs in an easterly direction. The
other bulldozer pushes the
piling with a force of 2150-lbs in
a northerly direction.
a.) What is the magnitude of
the resultant force upon
the piling?
(Draw a picture on the
whiteboard)
𝑟 2 = 9002 + 21502 = 5432500
𝑟 = 50√2173 ≈ 2330.77
b.) What is the direction of the
resulting force upon the
piling?
2150
tan 𝜃 =
900
2150
−1
𝜃 = 𝑡𝑎𝑛 (
) ≈ 67.29°
900
𝛼 = 90 − 67.29
= 22.71°𝐸𝑎𝑠𝑡 𝑜𝑓 𝑑𝑢𝑒 𝑁𝑜𝑟𝑡ℎ
EX: An airplane is flying due
west at a velocity of 100m/s. The
wind is blowing out of the south
at 5m/s. What is the magnitude
of the airplane’s resultant
velocity?
(Draw a picture on the
whiteboard)
𝑟 2 = 52 + 1002 = 10025
𝑟 = √10025 = 5√401 ≈ 100.12
EX: Radiology technicians Mary
Jones and Joe Rodriguez are
moving a patient on an MRI
machine cot. Ms. Jones is
pushing the cot with a force of
120 newtons at 55 degrees with
the horizontal while Mr.
Rodriguez is pulling the cot
with a force of 200 newtons at
40 degrees with the horizontal.
What is the magnitude of the
force exerted on the cot?
(Draw a picture on the
whiteboard)
𝑟 2 = 1202 + 2002 − 2(120)(200) cos 165°
𝑟 2 = 100764.44
𝑟 = 317.43
Upon completion of this lesson,
you should be able to:
1. Find the dot product of
vectors.
2. Determine if vectors are
orthogonal.
3. Solve story problems
involving vectors.
For more information, visit
http://www.mathsisfun.com/algebra/vector
s-dot-product.html
HW Pg.467 3-42 3rds, 71
Quiz over 6.3-6.4 tomorrow