M3 PRECALCULUS PACKET 2 FOR UNIT 6 – SECTION 6.4 & VECTORS IN 3D
SECTION 6.4 DOT PRODUCTS OF VECTORS
Part 1: Introduction to the Dot Product


 
The dot product of vectors u = u1 , u2 and v = v1 , v2 is written as u v ,
 
v u1v1 + u2v2 .
and it is found by u=


1. Find the dot product of vectors u and v :

u
a. =

2, −3 , v = 5,3


 


4i − 3 j , v =
2i + 5 j
b. u =
2. Is the dot product a vector or a scalar?
 
 
3. a. Does u v equal v u ?


b. Use u = u1 , u2 and v = v1 , v2 to prove or disprove your answer to #3a.

4. The zero vector 0,0 is shown by 0 , or it might be typed as a bold 0 in your textbook.
 
What does 0u equal?

5. =
Given u

v1 , v2 , and w = w1 , w2 :

=
u1 , u2 , v
 

 

 
 
a. Prove that u ( v + w ) = u v + u  w .
b. Prove that v v = v .
2
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M3 PRECALCULUS PACKET 2 FOR UNIT 6 – SECTION 6.4 & VECTORS IN 3D
Part 2: Angle Between Two Vectors
 
u v


The angle θ between two vectors u and v can be found by the formula cos θ =   .
u v
Here’s the proof:
(A)
(B)
(C)
θ
θ

(D)
θ
θ



(A) shows vectors u and v . In (B), draw the opposite of v at the end of u . The resultant
 
 
vector will be u − v . (C) shows the same vector u − v in a different location. (D) shows the
 
 
triangle formed by the vectors u , v , and u − v . Then by the Law of Cosines,
 2 2 2
 
u − v = u + v − 2 u v cos θ .


 
6. Let u = u1 , u2 and v = v1 , v2 . Rewrite u − v
2
2 
= u + v
2
in terms of u1 , u2 , v1 , and v2 ,
and substitute this into the equation above.
 
u v
7. Isolate cos θ in your equation from #6, and show that the other side matches   .
u v


 


4i − 3 j , v =
2i + 5 j . Round to
8. Use the formula to find the angle between the vectors u =
the nearest degree.
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M3 PRECALCULUS PACKET 2 FOR UNIT 6 – SECTION 6.4 & VECTORS IN 3D
Part 3: Orthogonal, Parallel, or Neither?
 
u v
9. Consider the formula for the angle between two vectors again: cos θ =   . If vectors
u v
are perpendicular, they are said to be orthogonal.


a. What is the value of θ when u and v are orthogonal?
 


b. What does that mean about the value of u v when u and v are orthogonal?
 


So vectors u and v are orthogonal when their dot product u v = ____.
10. How can you tell that all of the vectors below are parallel to each other?
−2, −3
4,6
2,3
1,1.5
20,30
11. Determine whether each pair of vectors is orthogonal, parallel, or neither.


 


3i − j , w =
6i − 2 j
a. v =


b. v =2, −1 , w =3,6


 


2i + 2 j , w =
i +2j
c. v =
3
M3 PRECALCULUS PACKET 2 FOR UNIT 6 – SECTION 6.4 & VECTORS IN 3D
GRAPHING IN 3D
z
Part 1: Graphing Points in Three Dimensions
When graphing in 3D, we use three axes, x, y, and z.
x
In this course, we’ll apply the right-hand rule for orienting the axes.
y
Position your right hand with the thumb pointing up, palm facing
toward you. Your fingers will wrap from the positive x-axis to the
positive y-axis. Your thumb represents the positive z-axis. Try this now.
The three axes separate space into 8 octants. The first octant is
bordered by the positive x-, y-, and z-axes. We will not cover the
numbering of the other octants in this course.
In 3D, there are 3 coordinate planes, xy-plane, the xz-plane,
and the yz-plane, each named after the 2 axes it contains.
Drawing perpendicular lines from a point to these planes forms
a rectangular prism, as shown in the diagram at right.
To graph a point in 3D, called an ordered triple, use dashed lines
to form a rectangle in the xy-plane where the x- and ycoordinates of the point meet. Then use a dashed vertical line
to show the z-coordinate extending upward or downward from
the corner of the rectangle. The diagram at right shows a point
that has positive x-, y-, and z-coordinates.
1. Graph each point:
a.
( −1, 2, −2 )
b.
( 3, −1, −1)
c. (1,0, 4 )
4
d.
( −3, −2,1)
M3 PRECALCULUS PACKET 2 FOR UNIT 6 – SECTION 6.4 & VECTORS IN 3D
Part 2: Distance and Midpoint in 3D
e
d
c
a
b
2. The diagram above shows a rectangular prism with opposite corners ( x1 , y1 , z1 ) and
( x2 , y2 , z2 ) . Follow the steps to derive a formula for the distance between these points.
a. Write expressions for the lengths of sides a, b, and e.
a = ________________
b = _________________
e = ___________________
b. Use the Pythagorean Theorem and sides a and b to find the length of side c.
c = ________________
c. Use the Pythagorean Theorem and sides c and e to find the length of side d.
d = ___________________________________________
Distance formula in 3D
5
M3 PRECALCULUS PACKET 2 FOR UNIT 6 – SECTION 6.4 & VECTORS IN 3D
3. Most formulas in 3D will just add a z-component to the formulas you already know. This
holds true for the midpoint formula in three dimensions. Write it below:


Midpoint between ( x1 , y1 , z1 ) and ( x2 , y2 , z2 ) = 
,
,



4. Find the distance and midpoint for each. Express distances in simplest radical form.
a.
( 0,1,3)
and (1, 4, −2 )
b.
( 7, −1, −2 ) and ( 3, −5, 4 )
Part 3: Vectors in 3D
As with many other formulas in three dimensions, we work with vectors in 3D mostly the same
as in 2D. Try these.
In #5-7, graph the vector with given initial and terminal points. Then find its component form.
5. Initial ( −6, 4, −2 ) , terminal (1, −1,3)
6. Initial ( −7,3,5 ) , terminal ( 0,0, 2 )
7. Initial ( −1, 2, −4 ) , terminal (1, 4, −4 )
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M3 PRECALCULUS PACKET 2 FOR UNIT 6 – SECTION 6.4 & VECTORS IN 3D
8. Find vector z = 3w – 2v + u, given that u = −1,3, 2 , v = 1, −2, −2 , and w = 5iˆ − 5kˆ .
9. Find the magnitude of v = 1, −2, 4 .
10. Find a unit vector in the direction of u = −3iˆ + 5 ˆj + 10kˆ .
 
11. Find u v for u = 2, −5,3 and v = 9i + 3j – k.
12. Find the angle θ between the vectors u = 0, 2, 2 and v = 3,0, −4 . Round your answer to
the nearest tenth of a degree.
13. Determine whether u =
neither.
3ˆ 1 ˆ
i − j + 2kˆ and v = 4iˆ + 10 ˆj + kˆ are orthogonal, parallel, or
4
2
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M3 PRECALCULUS PACKET 2 FOR UNIT 6 – SECTION 6.4 & VECTORS IN 3D
REVIEW FOR SECTIONS 6.3 AND 6.4
Part 1: Review of Formulas and Notation


Let u = u1 , u2 and v = v1 , v2 in component form.

1. What does u represent? What is its formula?

2. Express u in trigonometric form, given its direction angle θ .

3. What formula can be used to find the direction angle θ of u ?

4. Write a formula that finds the unit vector in the direction of u .

 


5. What vectors do i and j represent in two dimensions? What do i , j , and k represent in three
dimensions?



6. Express u as a linear combination of i and j .
 
7. Write a formula for u v .


8. Write a formula for the angle θ between u and v .


9. What does it mean for two vectors to be orthogonal? What should be true if u and v are orthogonal to each
other?


10. What should be true if u and v are parallel to each other?
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M3 PRECALCULUS PACKET 2 FOR UNIT 6 – SECTION 6.4 & VECTORS IN 3D
Part 2: Vector Operations
Round to 2 decimal places where necessary, final answers only. Denominators do not have to be rationalized.
11. Find the component form and magnitude of vector v with initial point P and terminal point Q:
a. P(−2,13), Q(5,4)
b. P(−3,−2), Q(−7,−8)
c. P(−2,2,5), Q(1,−3,0)
12. Let u= −2,3 and v= 1, −1 . Graph each, and write the resultant in component form:
a. u + v
b. u – v

c. 3u – 2v

13. Use the graph of u and v to sketch the resultant vector.
 


a. u − v
b. v − 2u
14. Find a unit vector in the direction of u.
a. u = −3, 7

15. Write each vector in trigonometric form.



b. u = 2i + 4 j

= 5i − 3 j
a. w

b. v =−12, −5

d. vector u with initial point (1, −5 ) and terminal point ( −8,3)
9
c. u = −2,1, 2



−3i − 7 j
c. z =
M3 PRECALCULUS PACKET 2 FOR UNIT 6 – SECTION 6.4 & VECTORS IN 3D
16. Find the component form of each vector with given magnitude and direction.
a.


v= 10, θ= 45°
v
b. =
17. Let u = −3, 4 , v = −1, −5 , and w = 0, 2 . Find:
1
,=
θ 200°
2
  
 
b. w( u + v )
a. u v






−5i + 9 j and v =
−3i − 6 j .
18. Find the angle (in degrees) between the two vectors, u =
19. Use vectors to find the measure of each angle in the triangle with vertices A(−1,0), B(2,1), and C(3,−3).
20. Which pairs of vectors are orthogonal?




3, −2 , w =
−1, 2
a. v =
1
2
−1, 2 , w =
0, −
c. v =



2, 3 , w =
−2,3
d. v =−


21. Find k so that u and v are orthogonal:


−2,0 , w =
0,5
b. v =


−3k ,5 , v =
2, −4
a. u =

 


3i 2 j , v =−
2i kj
b. u =+
10
M3 PRECALCULUS PACKET 2 FOR UNIT 6 – SECTION 6.4 & VECTORS IN 3D
Part 3: Vector Applications
Round to 2 decimal places where necessary, final answers only. Denominators do not have to be rationalized.
22. Three forces with magnitudes of 20 lbs, 60 lbs, and 90 lbs act on an object at angles 220 ° , 60 ° , and 140 °
respectively, with the positive x-axis. Find the magnitude and direction of the resultant of these forces.
23. A plane is heading due north at 560 mph. A 40-mph wind blows from the northeast at a bearing of N60 ° E.
Find the ground speed and direction of the plane.
24. Two forces with magnitudes of 200 and 300 act on an object at angles of 215 ° and 315 ° , respectively with
the positive x-axis. Find the direction and magnitude of the resultant of these forces.
25. A missile is launched with a velocity of 9500 ft/sec at an angle of 50 ° with the horizontal. Find the
horizontal and vertical components of the velocity.
11
M3 PRECALCULUS PACKET 2 FOR UNIT 6 – SECTION 6.4 & VECTORS IN 3D
26. A ball is thrown with an initial velocity of 65 feet per second at an angle of 43˚ with the horizontal. Find
the vertical and horizontal components of the velocity.
27. A 10,000 pound object is suspended on a wire tied to two poles. The angle between the horizontal and the
wire to the shorter pole is 22o. The angle between the horizontal and the wire to the taller pole is 49o. Find
the tension in each cable.
22o
49o
10,000 lbs
28. A boat travels 8.5 meters per second across a river. If the water flows downstream at a rate of 3.8 meters
per second, find the boat’s resultant velocity.
29. A 42-km/hr wind blows toward S55 ° W, while a plane heads N35 ° W at 152 km/hr. Find the resultant
velocity of the plane.
12
M3 PRECALCULUS PACKET 2 FOR UNIT 6 – SECTION 6.4 & VECTORS IN 3D
Selected Answers:
11. a. <7,-9>, 11.50
b. <-4,-6>, 7.21
c. <3,-5,-5>, 7.68
14. a.
7
−3
,
58 58
15. a.
34 cos ( −30.96° ) , 34 sin ( −30.96° )
b. 13cos 202.62°,13sin 202.62°
58 cos 246.8°, 58 sin 246.8°
d.
c.
16. a.
19.
22.
23.
25.
26.
28.
b.
5 2,5 2
55.30°,85.60°,39.10°
1 2
,
5 5
b.
c.
2 1 2
− , ,
3 3 3
−0.47, −0.17
20. B
145 cos138.37°, 145 sin138.37°
19. a. -17
21. a. k=-10/3
b. k=3
111.11 lbs, 119.23 °
541.11 mph, 93.67 °
24. 330.40, -81.59 °
Horizontal 6106.48 ft/sec, Vertical 7277.42 ft/sec
Horizontal 47.54 ft/sec, Vertical 44.33 ft/sec 30. 6939 lbs, 9806 lbs
9.31 m/sec
29. 157.7 km/hr
13
b. -2
18. 124.38°