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Draw the diagram of the 3d Cartesian plane.



Under what conditions may 2 vectors be parallel?
 vector u;
 vector v;
 c = some constant;
when u=cv

P = (2,1,4)
Q = (1,5,6)
R = (1,4,0)
S = (7,4,20)
Is vector PQ parallel to RS?
no.

What does a dot product do?
It multiplies vectors, turning them into scalars.

Give equation for dot product.
u*v = u_{1}v_{1} + u_{2}v_{2}



How do you find the angle between 2 vectors?
Invert cosine the dot product divided by the multiplied magnitude of both vectors.

u = <4,4>
v = <5,0>
What is the angle between u and v?
45 degrees

u = <5,1>
v = <4,2>
What is the angle between u and v?
142.1 degrees

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

What is another word for perpendicular?
orthogonal, norm

What's the z coordinate of any point on the xy plane?
0

What's the y coordinate of any point on the xz plane?
0

H = (3,3,3)
F = (3,8,3)
Whats the distance between H and F?
13.892

What's the distance formula?

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.

Find the coordinates of the midpoint of the line segment joining the points (4, 0, 10) and (4, 8, 20).
(4,4,5)

Find the standard equation of the sphere.
Endpoints of a diameter: (8, 0, 0), (0, 2, 0)
(x4)^{2}+(y1)^{2}+z^{2}=17

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 r^{2}=(xx1)^{2}+(yy1)^{2}+(zz_{1})^{2}


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

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

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

Find the scalar multiple of v and sketch its graph.
v = <1, 2, 2>
0v
<0,0,0>

Determine if the vectors is parallel to z. Initial_{z }(5, 4, 1) and terminal_{z} (−2, −4, 4).
7i + 6j + 2k
It is not.

Determine if the vectors is parallel to z. Initial_{z} (5, 4, 1) and terminal_{z} (−2, −4, 4).
−7i − 6j − 2k
It is not.

Determine if the vectors is parallel to z. Initial_{z} (5, 4, 1) and terminal_{z} (−2, −4, 4).
14i + 16j − 6k
It is.

Determine if the vectors is parallel to z. Initial_{z} (5, 4, 1) and terminal_{z} (−2, −4, 4).
−14i − 16j + 6k
It is.

Determine if the vectors is parallel to z. Initial_{z} (5, 4, 1) and terminal_{z} (−2, −4, 4).
i + 4j − 2k
It is not.

Determine if the vectors is parallel to z. Initial_{z} (5, 4, 1) and terminal_{z} (−2, −4, 4).
−i − 4j + 2k
It is not.

Use vectors to determine whether the points are collinear.
P = (0, −2, −5)
Q = (9, 4, 7)
R = (6, 2, 3)
They are collinear.

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.

Find the magnitude of vector v.
v = 8i + 5j + 9k

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/ >

Find the vector v with the given magnitude v and direction u.
Magnitude: 8
Direction: u = 2, 4, 0
< 8/ , 16/ , 0>

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

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

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

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

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

Find the angle θ between the vectors.
u = 2i − 5j + 2k
v = 4i − 4j + 2k
21.8 degrees

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

Determine whether u and v are orthogonal, parallel, or neither.
u = j + 9k
v = i − 3j − k
neither



P = (4,5,2)
Q = (1,6,7)
R = (1,2,5)
Are P, Q and R collinear?
No since no scalar exists.

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

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

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

u = <2,1>
v = <3,1>
w = <1,2>
Find 2(u*v)
14

u = <2,4>
v = <4,2>
Find angle between u and v.
90 degrees

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

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

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

u = <2,4>
v = <4,2>
w = <1,2>
Sketch.

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

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

u = <3,3,3>
v = <1,2,2>
Find angle between u and v.
54.7 degrees

u = <3,3,3>
w = <4,1,1>
Find angle between u and w.
35.3 degrees

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

u = <3,3,3>
v = <1,2,2>
w = <4,1,1>
Which 2 are orthogonal?
v and w

How do you determine if 2 vectors are orthogonal?
They are orthogonal if their dot product is 0

How do you find the dot product?
multiply the components of the vector by the components of the same dimension then add them.

