# 11.2, 11.3 - 3D Vectors

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 Author: NateGatsby ID: 194706 Filename: 11.2, 11.3 - 3D Vectors Updated: 2013-02-23 11:37:07 Tags: calc1 Folders: Description: unit 1 Show Answers:

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1. Draw the diagram of the 3-d Cartesian plane.
2. Plot (1,2,3).
3. P = (1,2,3)
Q = (-1, 4,6)

Graph vector PQ, provide magnitude and unit vector.
• magnitude =
• unit vector = <-2, 2/, 3/>
4. Under what conditions may 2 vectors be parallel?
• vector u;
• vector v;
• c = some constant;

when u=cv
5. P = (2,1,-4)
Q = (-1,5,6)
R = (1,4,0)
S = (7,-4,20)

Is vector PQ parallel to RS?
no.
6. What does a dot product do?
It multiplies vectors, turning them into scalars.
7. Give equation for dot product.
u*v = u1v1 + u2v2
8. <1,2>*<4,8>
20
9. <-1,4,7>*<5,2,0>
3
10. How do you find the angle between 2 vectors?
Invert cosine the dot product divided by the multiplied magnitude of both vectors.
11. u = <4,4>
v = <5,0>

What is the angle between u and v?
45 degrees
12. u = <-5,1>
v = <4,2>

What is the angle between u and v?
142.1 degrees
13. u = <0,4,2>
v = <2,3,0>

What is the angle between u and v?
41.9 degrees
14. What is another word for perpendicular?
orthogonal, norm
15. What's the z coordinate of any point on the xy plane?
0
16. What's the y coordinate of any point on the xz plane?
0
17. H = (-3,3,3)
F = (3,-8,-3)

Whats the distance between H and F?
13.892
18. What's the distance formula?
19. How do you find the middle of a vector?
Determine each value in the coordinate by finding the halfway point between the initial and terminal points.
20. Find the coordinates of the midpoint of the line segment joining the points (4, 0, -10) and (4, 8, 20).
(4,4,5)
21. Find the standard equation of the sphere.

Endpoints of a diameter: (8, 0, 0), (0, 2, 0)
(x-4)2+(y-1)2+z2=17
22. How do you find the equation of a sphere based off the coordinates of the end points?
First find the center. This is done by find the halfway point between each value in the coordinates. Then you plug in the coordiante of the center to the equation r2=(x-x1)2+(y-y1)2+(z-z1)2
23. Find the component form and magnitude of the vector v with the given initial and terminal points. Then find a unit vector in the direction of v.

Initial Point: (4, 8, 0)
Terminal Point:(4, 1, 2)
• v = <0,-7,2>
• magnitude  =
• unit vector = < 0, -7/, 2/ >
24. Find the scalar multiple of v and sketch its graph.
v = <1, 2, 2>

2v
<2,4,4>
25. Find the scalar multiple of v and sketch its graph.
v = <1, 2, 2>

-v
<-1,-2,-2>
26. Find the scalar multiple of v and sketch its graph.
v = <1, 2, 2>

(3/2)v
<3/2,3,3>
27. Find the scalar multiple of v and sketch its graph.
v = <1, 2, 2>

0v
<0,0,0>
28. Determine if the vectors is  parallel to z. Initialz (5, 4, 1) and terminalz (−2, −4, 4).

7i + 6j + 2k
It is not.
29. Determine if the vectors is  parallel to z. Initialz (5, 4, 1) and terminalz (−2, −4, 4).

−7i − 6j − 2k
It is not.
30. Determine if the vectors is  parallel to z. Initialz (5, 4, 1) and terminalz (−2, −4, 4).

14i + 16j − 6k
It is.
31. Determine if the vectors is  parallel to z. Initialz (5, 4, 1) and terminalz (−2, −4, 4).

−14i − 16j + 6k
It is.
32. Determine if the vectors is  parallel to z. Initialz (5, 4, 1) and terminalz (−2, −4, 4).

i + 4j − 2k
It is not.
33. Determine if the vectors is  parallel to z. Initialz (5, 4, 1) and terminalz (−2, −4, 4).

−i − 4j + 2k
It is not.
34. Use vectors to determine whether the points are collinear.

P = (0, −2, −5)
Q = (9, 4, 7)
R = (6, 2, 3)
They are collinear.
35. How do you determine if points are collinear?
You find the vectors between a base point and 2 other points. If the vectors are parallel (ie have a constant variable linking them) then they are collinear.
36. Find the magnitude of vector v.

v = -8i + 5j + 9k
37. Find a unit vector in the direction of v and in the direction opposite of v.

v = < 7, 1, 5 >
in the direction: < 7/ , 1/ , 5/ >

opposite direction: < -7/ , -1/ , -5/ >
38. Find the vector v with the given magnitude v and direction u.

Magnitude: 8
Direction: u = -2, 4, 0
< -8/ , 16/ , 0>
39. u = <6, 14>,    v = <−2, 3>

(a)    u · v
(b)    u · u
(c)    ||u||2
(d)    (u · v)v
(e)    u · (2v)
• a) 30
• b) 232
• c) 232
• d) <-60,90>
• e) 60
40. Find the angle θ between the vectors.

u = 3i + j, v = −2i + 4j
98.1 degrees
41. Find the angle θ between the vectors.

u=cos(/6)i+sin(/6)j
v=cos(3/4)i+sin(3/4)j
105 degrees
42. Find the angle θ between the vectors.

u = <4, 4>
v = <8, -8 >
90 degrees
43. Find the angle θ between the vectors.

u = 3i + 4j + k
v = 4i − 3j
90 degrees
44. Find the angle θ between the vectors.

u = 2i − 5j + 2k
v = 4i − 4j + 2k
21.8 degrees
45. Determine whether u and v are orthogonal, parallel, or neither.

u = <cos(θ), sin(θ), 1 >
v =< sin(θ), −cos(θ), 0>
orthogonal
46. Determine whether u and v are orthogonal, parallel, or neither.

u = j + 9k
v = i − 3j − k
neither
47. P = (4,-5,2)
Q = (-1,6,7)

Find mid point, unit vector and magnitude of vector PQ as well as the component form in linear combination.
• mid point: <3/2 , 1/2 , 9/2 >
• unit vector: < -5/ , 11/, 5/ >
• magnitude:
• -5i+11j+5k
48. P = (4,-5,2)
R = (1,-2,5)

Find mid point, unit vector and magnitude of vector PR as well as the component form in linear combination.
• mid point: <5/2 , -7/2 , 7/2 >
• unit vector: < -3/ , 3/, 3/ >
• magnitude:
• -3i+3j+3k
49. P = (4,-5,2)
Q = (-1,6,7)
R = (1,-2,5)

Are P, Q and R collinear?
No since no scalar exists.
50. u = <2,-1>
v = <-3,1>
w = <1,-2>

Find u*v.
-7
51. u = <2,-1>
v = <-3,1>
w = <1,-2>

Find v(u*w)
<-12,4>
52. u = <2,-1>
v = <-3,1>
w = <1,-2>

2(u+v)
-2i
53. u = <2,-1>
v = <-3,1>
w = <1,-2>

Find 2(u*v)
-14
54. u = <-2,4>
v = <-4,-2>

Find angle between u and v.
90 degrees
55. u = <-2,4>
w = <1,-2>

Find angle between u and w.
53.1 degrees
56. v = <-4,-2>
w = <1,-2>

Find angle between w and v.
143.1 degrees
57. u = <-2,4>
v = <-4,-2>
w = <1,-2>

Which 2 vectors are orthogonal?
v and u
58. u = <-2,4>
v = <-4,-2>
w = <1,-2>

Sketch.
59. u = <3,4,12>
v = <-4,3,0>
w = <4,5,-3>

Find u*v.
0
60. u = <3,4,12>
v = <-4,3,0>
w = <4,5,-3>

Find u*w.
-4
61. u = <3,-3,3>
v = <-1,-2,2>

Find angle between u and v.
54.7 degrees
62. u = <3,-3,3>
w = <4,-1,1>

Find angle between u and w.
35.3 degrees
63. v = <-1,-2,2>
w = <4,-1,1>

Find angle between w and v.
90 degrees
64. u = <3,-3,3>
v = <-1,-2,2>
w = <4,-1,1>

Which 2 are orthogonal?
v and w
65. How do you determine if 2 vectors are orthogonal?
They are orthogonal if their dot product is 0
66. How do you find the dot product?
multiply the components of the vector by the components of the same dimension then add them.

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