Alg1131W4T1 - Lines and planes
Friday, 23 March 2018
1:29 PM
Created with Microsoft OneNote 2016.
![Untitled picture.png Machine generated alternative text:
In two dimensions, the length ot a vector
denoted Ivl, is naturally defined as the square root ot the quadrance Q (v) a2 + b2, in other words
It v = AB then the length of v is also the distance between the points A and B, and we write
b
2
So the length of the vector u =
—1
Ivl= AB .
O . And the distance between the points A = [—2, 2] and B = [—3, —2] is dist(A,B)
Note: to write the square root of 2 in Maple we would use sqrt (2)
In three dimensions the story is similar, where the length of the vector
b
denoted Ivl, is defined as the square root ot the quadrance Q (v) a2 + b2 + c2 in other words
So the length of w
1
3
is
sqrt(1 0)
O and the distance between the points C = and D = [1, 8,3] is
sqrt(26)
An important related idea is that ot the dot product ot two vectors. In three-dimensional space, it
a2
bl
we define
So for example
and
- ald2+b1b2+c102.
—1
5
3
2
1
4
15
O , and
—1
5
3
—1
5
3
VI
= 35.
We observe that in general v . v is just the quadrance Q(v)
Untitled picture.png Machine generated alternative text:
Consider the plane y = 1 in three-dimensional space, together witn the unit cube with vertices O, A, B, C, D, E, F, G as shown below
Geometrically, a vector u is parallel to a plane it there exist points P and Q on the plane such that u = PQ.
For example OE is parallel to the plane y = 1 because OE = CF where C and F are öotn on the plane.
Geometrically this means that OEFC is a parallelogram.
i) Which of these vectors are parallel to the plane y = 1 ? [select all that apply]
OD
D OC
D GO.
ii) The vector DC is parallel to the planes [select all that apply)
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𝑣=𝑄𝑣=𝑥1−𝑥22+𝑦1−𝑦22+𝑧1−𝑧22+…
𝑣=𝑄𝑣=𝑎2+𝑏2+𝑐2+…
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−1×2+5×1+3×4=15
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![Untitled picture.png Machine generated alternative text:
Consider the plane y = 1 in three-dimensional space, together witn the unit cube with vertices O, A, B, C, D, E, F, G as shown below
Geometrically, a vector u is parallel to a plane it there exist points P and Q on the plane such that u = PQ.
For example OE is parallel to the plane y = 1 because OE = CF where C and F are öotn on the plane.
Geometrically this means that OEFC is a parallelogram.
i) Which of these vectors are parallel to the plane y = 1 ? [select all that apply]
OD
D OC
D GO.
ii) The vector DC is parallel to the planes [select all that apply)
Untitled picture.png Machine generated alternative text:
Here is another view ot the same unit cube. We see that the diagonal lines OF, EC, AG and DB ottne unit cube meet at the point P in the center.
i) The point Pis
ii) The vectors PA, PB and PC are:
PA
iii) Evaluate the following dot products:
O and BC. BE
An important use of the dot product is defining perpendicularity:
So we can conclude that
u I v precisely when u • v =
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](Alg1131W4T1%20-%20Lines%20and%20planes_files/image002.png)
![Untitled picture.png Machine generated alternative text:
3
2
o
0
05
2
3
4
5
G
The dot product allows us to also define and calculate the angle 9 between vectors v and w by the formula
v.w= Ivllwlcos(9).
Here is our unit cube again'
i) The cosine ot the angle Z ADC is 112
ii) The triangle AADCis equilateral
iii) The angle Z BEG is 60
O degrees.
iv) The triangle AEBG is equilateral
v) The two triangles A ADC and AEBG are:
congruent
parallel
similar, but not congruent
meeting at the center of the cube
Untitled picture.png Machine generated alternative text:
The dot product is defined between any two vectors VI and v2 in n-dimensional space, and gives us the useful relation
where 9 is the angle between VI and v2_
In two dimensions it v =
al
and v 2 ¯
a2
then the dot product is
Consider the vectors
¯ IVI IV'21 cos(9),
VI = ala2 +b1b2-
2
0
i) Then a • b
2
2
3
3
4
3
and b
1
1
2
x
5
ii) The cosine ot the angle 9 between a and b is cos(9)
iii) The angle between a and b is, to the nearest degree, 46
iv) The cosine ot the angle between a and BA is
1/sqrt(2)
v) The angle between a and BA is, to the nearest degree,
vi) The triangle formed by O = [0,0], A = [3,11 and B
a right triangle
obtuse
isosceles
equilateral
46
[1, 2] is a bit special, since it is: [select all that apply]
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](Alg1131W4T1%20-%20Lines%20and%20planes_files/image004.png)
![Untitled picture.png Machine generated alternative text:
The dot product is defined between any two vectors VI and v2 in n-dimensional space, and gives us the useful relation
where 9 is the angle between VI and v2_
In two dimensions it v =
al
and v 2 ¯
a2
then the dot product is
Consider the vectors
¯ IVI IV'21 cos(9),
VI = ala2 +b1b2-
2
0
i) Then a • b
2
2
3
3
4
3
and b
1
1
2
x
5
ii) The cosine ot the angle 9 between a and b is cos(9)
iii) The angle between a and b is, to the nearest degree, 46
iv) The cosine ot the angle between a and BA is
1/sqrt(2)
v) The angle between a and BA is, to the nearest degree,
vi) The triangle formed by O = [0,0], A = [3,11 and B
a right triangle
obtuse
isosceles
equilateral
46
[1, 2] is a bit special, since it is: [select all that apply]
Untitled picture.png Machine generated alternative text:
cos 0
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](Alg1131W4T1%20-%20Lines%20and%20planes_files/image005.png)