Outline
Week 2: Determinants, Cross Products, Lines, and Planes
Course Notes: 2.4-2.5
Goals: Introduce determinants and cross products, computationally and
with geometric interpretations. Lines and planes.
Matrices!


8 15 −4
 9 −4 7 


6
1
1 
−5 −3 0
Determinants in Two and Three Dimensions
a1 a2
det
= a1 b2 − a2 b1
b1 b2
Determinants in Two and Three Dimensions
a1 a2
det
= a1 b2 − a2 b1
b1 b2

a1 a2 a3
b
b
b
b
b
b
2
3
1
3
1
2
− a2 det
+ a3 det
det b1 b2 b3  = a1 det
c2 c3
c1 c3
c1 c2
c1 c2 c3

Determinants in Two and Three Dimensions
a1 a2
det
= a1 b2 − a2 b1
b1 b2

a1 a2 a3
b
b
b
b
b
b
2
3
1
3
1
2
− a2 det
+ a3 det
det b1 b2 b3  = a1 det
c2 c3
c1 c3
c1 c2
c1 c2 c3

Tricky way: ONLY in three dimensions:


a1 a2 a3
det b1 b2 b3  = a1 b2 c3 + a2 b3 c1 + a3 b1 c2 − a3 b2 c1 − a2 b1 c3 − a1 b3 c2
c1 c2 c3
Determinants in Two and Three Dimensions
a1 a2
det
= a1 b2 − a2 b1
b1 b2

a1 a2 a3
b
b
b
b
b
b
2
3
1
3
1
2
− a2 det
+ a3 det
det b1 b2 b3  = a1 det
c2 c3
c1 c3
c1 c2
c1 c2 c3

Tricky way: ONLY in three dimensions:


a1 a2 a3
det b1 b2 b3  = a1 b2 c3 + a2 b3 c1 + a3 b1 c2 − a3 b2 c1 − a2 b1 c3 − a1 b3 c2
c1 c2 c3
Determinants in Two and Three Dimensions
a1 a2
det
= a1 b2 − a2 b1
b1 b2

a1 a2 a3
b
b
b
b
b
b
2
3
1
3
1
2
− a2 det
+ a3 det
det b1 b2 b3  = a1 det
c2 c3
c1 c3
c1 c2
c1 c2 c3

Tricky way: ONLY in three dimensions:


a1 a2 a3
det b1 b2 b3  = a1 b2 c3 + a2 b3 c1 + a3 b1 c2 − a3 b2 c1 − a2 b1 c3 − a1 b3 c2
c1 c2 c3
Determinants in Two and Three Dimensions
a1 a2
det
= a1 b2 − a2 b1
b1 b2

a1 a2 a3
b
b
b
b
b
b
2
3
1
3
1
2
− a2 det
+ a3 det
det b1 b2 b3  = a1 det
c2 c3
c1 c3
c1 c2
c1 c2 c3

Tricky way: ONLY in three dimensions:


a1 a2 a3
det b1 b2 b3  = a1 b2 c3 + a2 b3 c1 + a3 b1 c2 − a3 b2 c1 − a2 b1 c3 − a1 b3 c2
c1 c2 c3
Determinants in Two and Three Dimensions
a1 a2
det
= a1 b2 − a2 b1
b1 b2

a1 a2 a3
b
b
b
b
b
b
2
3
1
3
1
2
− a2 det
+ a3 det
det b1 b2 b3  = a1 det
c2 c3
c1 c3
c1 c2
c1 c2 c3

Tricky way: ONLY in three dimensions:


a1 a2 a3
det b1 b2 b3  = a1 b2 c3 + a2 b3 c1 + a3 b1 c2 − a3 b2 c1 − a2 b1 c3 − a1 b3 c2
c1 c2 c3
Determinants in Two and Three Dimensions
a1 a2
det
= a1 b2 − a2 b1
b1 b2

a1 a2 a3
b
b
b
b
b
b
2
3
1
3
1
2
− a2 det
+ a3 det
det b1 b2 b3  = a1 det
c2 c3
c1 c3
c1 c2
c1 c2 c3

Tricky way: ONLY in three dimensions:


a1 a2 a3
det b1 b2 b3  = a1 b2 c3 + a2 b3 c1 + a3 b1 c2 − a3 b2 c1 − a2 b1 c3 − a1 b3 c2
c1 c2 c3
Determinants in Two and Three Dimensions
a1 a2
det
= a1 b2 − a2 b1
b1 b2

a1 a2 a3
b
b
b
b
b
b
2
3
1
3
1
2
− a2 det
+ a3 det
det b1 b2 b3  = a1 det
c2 c3
c1 c3
c1 c2
c1 c2 c3

Tricky way: ONLY in three dimensions:


a1 a2 a3
det b1 b2 b3  = a1 b2 c3 + a2 b3 c1 + a3 b1 c2 − a3 b2 c1 − a2 b1 c3 − a1 b3 c2
c1 c2 c3
1 3
det
2 5
1 3
det
2 5
det
−2 8
3 5
1 3
det
2 5
det
−2 8
3 5


3 2 5
det 5 7 3
2 1 3
1 3
det
2 5
det
−2 8
3 5


3 2 5
det 5 7 3
2 1 3


2 4 8
det 3 5 7
1 10 5
Geometric Interpretation: Determinant in Two Dimensions
a1 a2
det
= a1 b2 − a2 b1
b1 b2
Geometric Interpretation: Determinant in Two Dimensions
a1 a2
det
= a1 b2 − a2 b1
b1 b2
−a2
b
=
· 1
a1
b2
Geometric Interpretation: Determinant in Two Dimensions
a1 a2
det
= a1 b2 − a2 b1
b1 b2
−a2
b
=
· 1
a1
b2
y
−a2
a1
a1
a2
x
Geometric Interpretation: Determinant in Two Dimensions
a1 a2
det
= a1 b2 − a2 b1
b1 b2
−a2
b
=
· 1
a1
b2
If det
then
a1 a2
= 0,
b1 b2
Geometric Interpretation: Determinant in Two Dimensions
a1 a2
det
= a1 b2 − a2 b1
b1 b2
−a2
b
=
· 1
a1
b2
a a
If det 1 2 = 0,
b b
1 2
−a2
b
is orthogonal to 1 ,
then
a1
b2
Geometric Interpretation: Determinant in Two Dimensions
a1 a2
det
= a1 b2 − a2 b1
b1 b2
−a2
b
=
· 1
a1
b2
a a
If det 1 2 = 0,
b b
1 2
−a2
b
is orthogonal to 1 ,
then
a1
b2
a
b
so 1 and 1 are
a2
b2
Geometric Interpretation: Determinant in Two Dimensions
a1 a2
det
= a1 b2 − a2 b1
b1 b2
−a2
b
=
· 1
a1
b2
a a
If det 1 2 = 0,
b b
1 2
−a2
b
is orthogonal to 1 ,
then
a1
b2
a
b
so 1 and 1 are scalar multiples of one another (parallel).
a2
b2
Zero Determinants
Are the following determinants zero, or nonzero?
1
2
det
−4 −8
Zero Determinants
Are the following determinants zero, or nonzero?
1
2
det
=0
−4 −8
Zero Determinants
Are the following determinants zero, or nonzero?
1
2
det
=0
−4 −8
1 2
det
4 6
Zero Determinants
Are the following determinants zero, or nonzero?
1
2
det
=0
−4 −8
1 2
6 0
=
det
4 6
Zero Determinants
Are the following determinants zero, or nonzero?
1
2
det
=0
−4 −8
1 2
6 0
=
det
4 6
a
a2
det 1
5a1 5a2
Zero Determinants
Are the following determinants zero, or nonzero?
1
2
det
=0
−4 −8
1 2
6 0
=
det
4 6
a
a2
det 1
5a1 5a2
=0
Zero Determinants
Are the following determinants zero, or nonzero?
1
2
det
=0
−4 −8
1 2
6 0
=
det
4 6
a
a2
det 1
5a1 5a2
a1 a2
det
0 0
=0
Zero Determinants
Are the following determinants zero, or nonzero?
1
2
det
=0
−4 −8
1 2
6 0
=
det
4 6
a
a2
det 1
5a1 5a2
=0
a1 a2
=0
det
0 0
More Geometric Interpretation in Two Dimensions
det
a1 a2
−a2
b
=
· 1 = â · b
b1 b2
a1
b2
b
â
θ̂
a
θ
More Geometric Interpretation in Two Dimensions
det
a1 a2
−a2
b
=
· 1 = â · b
b1 b2
a1
b2
= kâkkbk cos(θ̂)
b
â
θ̂
a
θ
More Geometric Interpretation in Two Dimensions
det
a1 a2
−a2
b
=
· 1 = â · b
b1 b2
a1
b2
= kâkkbk cos(θ̂)
= kâkkbk cos(π/2 − θ)
b
â
θ̂
a
θ
More Geometric Interpretation in Two Dimensions
det
a1 a2
−a2
b
=
· 1 = â · b
b1 b2
a1
b2
= kâkkbk cos(θ̂)
= kâkkbk cos(π/2 − θ)
= kakkbk sin(θ)
b
â
θ̂
a
θ
More Geometric Interpretation in Two Dimensions
det
a1 a2
−a2
b
=
· 1 = â · b
b1 b2
a1
b2
= kâkkbk cos(θ̂)
= kâkkbk cos(π/2 − θ)
= kakkbk sin(θ)
b
a
θ
More Geometric Interpretation in Two Dimensions
det
a1 a2
−a2
b
=
· 1 = â · b
b1 b2
a1
b2
= kâkkbk cos(θ̂)
= kâkkbk cos(π/2 − θ)
= kakkbk sin(θ)
b
a
θ
More Geometric Interpretation in Two Dimensions
det
a1 a2
−a2
b
=
· 1 = â · b
b1 b2
a1
b2
= kâkkbk cos(θ̂)
= kâkkbk cos(π/2 − θ)
= kakkbk sin(θ)
kbk
b
sin θ
a
θ
More Geometric Interpretation in Two Dimensions
det
a1 a2
−a2
b
=
· 1 = â · b
b1 b2
a1
b2
= kâkkbk cos(θ̂)
= kâkkbk cos(π/2 − θ)
= kakkbk sin(θ)
kbk
b
sin θ
a
θ
More Geometric Interpretation in Two Dimensions
det
a1 a2
−a2
b
=
· 1 = â · b
b1 b2
a1
b2
= kâkkbk cos(θ̂)
= kâkkbk cos(π/2 − θ)
= kakkbk sin(θ) = shaded area
kbk
b
sin θ
a
θ
In general:
det a1 a2 = area of parallelogram spanned by a1 and b1
a2
b2
b1 b2 In general:
det a1 a2 = area of parallelogram spanned by a1 and b1
a2
b2
b1 b2 2
Example: Find the area of the parallelogram with one side given by
6
−3
and the other side
.
4
In general:
det a1 a2 = area of parallelogram spanned by a1 and b1
a2
b2
b1 b2 2
Example: Find the area of the parallelogram with one side given by
6
−3
and the other side
.
4
2 6
det
= (2)(4) − (6)(−3)
−3 4
= 8 + 18 = 26
In general:
det a1 a2 = area of parallelogram spanned by a1 and b1
a2
b2
b1 b2 2
Example: Find the area of the parallelogram with one side given by
6
−3
and the other side
.
4
2 6
det
= (2)(4) − (6)(−3)
−3 4
= 8 + 18 = 26
Silly Example: Find the area of the rectangle with corners (0, 0), (x, 0),
(0, y ), and (x, y ).
Cross Product
ONLY defined in threedimensions.

a1
a = a2 
a3
 
b1

b = b2 
b3
Cross Product
ONLY defined in threedimensions.

 
a1
b1



a = a2
b = b2 
a3
b3


a2 b3 − a3 b2

a × b = a3 b1 − a1 b3 
a1 b2 − a2 b1
Cross Product
ONLY defined in threedimensions.

 
a1
b1



a = a2
b = b2 
a3
b3


a2 b3 − a3 b2

a × b = a3 b1 − a1 b3 
a1 b2 − a2 b1
Mnemonic:


i
j k
a
a
a
a
a
a
2
3
1
3
1
2
det a1 a2 a3  = i det
− j det
+ k det
b2 b3
b1 b3
b1 b2
b1 b2 b3
= i(a2 b3 − a3 b2 ) − j(a1 b3 − a3 b1 ) + k(a1 b2 − a2 b1 )


a2 b3 − a3 b2
= a3 b1 − a1 b3 
a1 b2 − a2 b1
Practice
   
1
0
2 ×  1 
3
−1

  
0
1
 1  × 2
−1
3
Practice
   
1
0
2 ×  1 
3
−1
Not commutative!

  
0
1
 1  × 2
−1
3
Geometric Interpretation
1. a × b is orthogonal to a and to b .
Geometric Interpretation
1. a × b is orthogonal to a and to b .
Verify:

  
a2 b3 − a3 b2
a1
a3 b1 − a1 b3  · a2  = 0
a1 b2 − a2 b1
a3
Geometric Interpretation
1. a × b is orthogonal to a and to b .
Verify:

  
a2 b3 − a3 b2
a1
a3 b1 − a1 b3  · a2  = 0
a1 b2 − a2 b1
a3
2. ka × bk = kakkbk sin θ,
where θ is the angle between a and b , 0 ≤ θ ≤ π.
Geometric Interpretation
1. a × b is orthogonal to a and to b .
Verify:

  
a2 b3 − a3 b2
a1
a3 b1 − a1 b3  · a2  = 0
a1 b2 − a2 b1
a3
2. ka × bk = kakkbk sin θ,
where θ is the angle between a and b , 0 ≤ θ ≤ π.
Thus, sin θ is positive, and ka × bk is the area of the parallelogram
spanned by a and b .
Geometric Interpretation
1. a × b is orthogonal to a and to b .
Verify:

  
a2 b3 − a3 b2
a1
a3 b1 − a1 b3  · a2  = 0
a1 b2 − a2 b1
a3
2. ka × bk = kakkbk sin θ,
where θ is the angle between a and b , 0 ≤ θ ≤ π.
Thus, sin θ is positive, and ka × bk is the area of the parallelogram
spanned by a and b .
3. The vectors a , b , and a × b obey the right hand rule. That is, if you
curl your fingers towards your palm from a to b , your thumb points
in the direction of a × b.
b
a
b
a
b
a
kak
b
proja b
a
kak
b
b − proja b
proja b
a
kak
b
b − proja b
a
kak
A = (base)(height) = kakkb − proja bk
ka × bk = area of parallelogram
A2 = kak2 kb − proja bk2
ka × bk = area of parallelogram
A2 = kak2 kb − proja bk2
2
a·b
b
−
= kak2 a
kak2
ka × bk = area of parallelogram
A2 = kak2 kb − proja bk2
2
a·b
b
−
= kak2 a
kak2
!
2
a
·
b
a
·
b
= kak2 b · b − 2(b · a)
kak2
+
kak2
kak2
(a · b)2
= kak2 kbk2 −
kak2
ka × bk = area of parallelogram
A2 = kak2 kb − proja bk2
2
a·b
b
−
= kak2 a
kak2
!
2
a
·
b
a
·
b
= kak2 b · b − 2(b · a)
kak2
+
kak2
kak2
(a · b)2
= kak2 kbk2 −
kak2
= kak2 kbk2 − (a · b)2
= (a12 + a22 + a32 )2 (b12 + b22 + b32 )2 − (a1 b1 + a2 b2 + a3 b3 )2
= · · · = (a2 b3 − a3 b2 )2 + (a3 b1 − a1 b3 )2 + (a1 b2 − a2 b1 )2
= ka × bk2
Find the Area of the Parallelograms
 
 
1
3



1
Find the area of the parallelogram spanned by
and 1 .
1
−2
1
4
Find the area of the parallelogram spanned by
and
.
2
3
Find the Area of the Parallelograms
 
 
1
3



1
Find the area of the parallelogram spanned by
and 1 .
1
−2
1
4
Find the area of the parallelogram spanned by
and
.
2
3
Can you do that with a cross product, by imagining these vectors in R3 ?
Suppose a plane contains the points P1 (3, 2, 2), P2 (2, 2, 1), and
P3 (1, 1, 1). Find a normal vector to the plane. That is, find a vector that
is perpendicular to every line on the plane.
Properties of Cross Product
1. a × b =
b×a
Properties of Cross Product
1. a × b = −b × a
Properties of Cross Product
1. a × b = −b × a
2. a × (b × c) = (c · a)b − (b · a)c
Properties of Cross Product
1. a × b = −b × a
2. a × (b × c) = (c · a)b − (b · a)c
https://proofwiki.org/wiki/Lagrange’s_Formula
Properties of Cross Product
1. a × b = −b × a
2. a × (b × c) = (c · a)b − (b · a)c
3. s(a × b) = (sa) × b = a × (sb)
Properties of Cross Product
1. a × b = −b × a
2. a × (b × c) = (c · a)b − (b · a)c
3. s(a × b) = (sa) × b = a × (sb)
b
a
Properties of Cross Product
1. a × b = −b × a
2. a × (b × c) = (c · a)b − (b · a)c
3. s(a × b) = (sa) × b = a × (sb)
b
a
Properties of Cross Product
1. a × b = −b × a
2. a × (b × c) = (c · a)b − (b · a)c
3. s(a × b) = (sa) × b = a × (sb)
b
a
3a
Properties of Cross Product
1. a × b = −b × a
2. a × (b × c) = (c · a)b − (b · a)c
3. s(a × b) = (sa) × b = a × (sb)
b
a
3a
Properties of Cross Product
1. a × b = −b × a
2. a × (b × c) = (c · a)b − (b · a)c
3. s(a × b) = (sa) × b = a × (sb)
b
a
4. a × (b + c) = a × b + a × c
3a
Properties of Cross Product
1. a × b = −b × a
2. a × (b × c) = (c · a)b − (b · a)c
3. s(a × b) = (sa) × b = a × (sb)
b
a
4. a × (b + c) = a × b + a × c
5. a · (b × c) = (a × b) · c
3a
Properties of Cross Product
1. a × b = −b × a
2. a × (b × c) = (c · a)b − (b · a)c
3. s(a × b) = (sa) × b = a × (sb)
b
a
3a
4. a × (b + c) = a × b + a × c
5. a · (b × c) = (a × b) · c
Is it also true that (a · b) × c = a × (b · c)?
Properties of Cross Product
1. a × b = −b × a
2. a × (b × c) = (c · a)b − (b · a)c
3. s(a × b) = (sa) × b = a × (sb)
b
a
3a
4. a × (b + c) = a × b + a × c
5. a · (b × c) = (a × b) · c
“triple product”
Parallelograms
Parallelograms
Parallelograms
Parallelograms
Parallelograms
Area: (base)×(height)
Parallelepipeds
Parallelepipeds
Parallelepipeds
Parallelepipeds
Parallelepipeds
Volume: (area of base)×(height)
Triple Product: a · (b × c)
a
c
b
Triple Product: a · (b × c)
a
c
b
Triple Product: a · (b × c)
a
c
b
Triple Product: a · (b × c)
a
c
b
Area of base: kb × ck
Triple Product: a · (b × c)
b×c
θ
a
c
b
Area of base: kb × ck
Triple Product: a · (b × c)
kak| cos θ|
b×c
θ
a
c
b
Area of base: kb × ck
Triple Product: a · (b × c)
kak| cos θ|
b×c
θ
a
c
b
Area of base: kb × ck
Height of parallelepiped: kak| cos θ|
Triple Product: a · (b × c)
kak| cos θ|
b×c
θ
a
c
b
Area of base: kb × ck
Height of parallelepiped: kak| cos θ|
Volume of parallelepiped:
(area of base)(height)= kakkb × ck| cos θ| = |a · (b × c)|
Calculating the Triple Product
a · (b × c) =
Calculating the Triple Product


i
j k
a · (b × c) = a · det b1 b2 b3 
c1 c2 c3
Calculating the Triple Product
b2
 det c2


k

b


b3 = a · − det 1

c1
c3

b
det 1
c1


i
j

a · (b × c) = a · det b1 b2
c1 c2

b3
c3 

b3 


c
3 
b2 
c2
Calculating the Triple Product

b2 b3
 det c2 c3 




i
j k

b1 b3 




a · (b × c) = a · det b1 b2 b3 = a · − det

c
c
1
3


c1 c2 c3

b b 
det 1 2
c1 c2
b2 b3
b1 b3
b1 b2
= a1 det
− a2 det
+ a3 det
c2 c3
c1 c3
c1 c2

Calculating the Triple Product

b2 b3
 det c2 c3 




i
j k

b1 b3 




a · (b × c) = a · det b1 b2 b3 = a · − det

c
c
1
3


c1 c2 c3

b b 
det 1 2
c1 c2
b2 b3
b1 b3
b1 b2
= a1 det
− a2 det
+ a3 det
c2 c3
c1 c3
c1 c2


a1 a2 a3

= det b1 b2 b3 
c1 c2 c3

   
0
1
Find the volume of the parallelepiped spanned by 0, −1, and
1
1
 
−1
−1.
1
   
0
1
Find the volume of the parallelepiped spanned by 0, −1, and
1
1
 
−1
−1.
1
For positive a, b, and c, find the determinant and interpret it as a volume:


a 0 0
det 0 b 0
0 0 c
   
0
1
Find the volume of the parallelepiped spanned by 0, −1, and
1
1
 
−1
−1.
1
For positive a, b, and c, find the determinant and interpret it as a volume:


a 0 0
det 0 b 0
0 0 c
Calculate and explain geometrically:


2 0 3
det  8 1 7 
20 3 15
Right-Hand Rule
Predict the following cross products without using the cross-product
calculation. Draw your results. Check using the cross-product calculation.
   
2
0
0 × 0
0
7
   
0
0
0 × 2
7
0
   
−2
0
 0  × 7
0
0
Given any 3-dimensional vector a, is there a simple expression for a × a?
Given any 3-dimensional vector a, is there a simple expression for a × a?
What about (sa) × a for a scalar s?
Given any 3-dimensional vector a, is there a simple expression for a × a?
What about (sa) × a for a scalar s?
What about a · (a × b)?
Given any 3-dimensional vector a, is there a simple expression for a × a?
What about (sa) × a for a scalar s?
What about a · (a × b)?
Consider a × (b × c). Will this vector be in the same plane as b and c , or
in an orthogonal plane?
Given any 3-dimensional vector a, is there a simple expression for a × a?
What about (sa) × a for a scalar s?
What about a · (a × b)?
Consider a × (b × c). Will this vector be in the same plane as b and c , or
in an orthogonal plane?
Notice a × (b × c) = (c · a)b − (b · a)c: a linear combination of b and c .
Parametric Equations of Lines
x2
x1
Parametric Equations of Lines
x2
a
x1
Parametric Equations of Lines
x2
a
x1
Parametric Equations of Lines
x2
a
x1
x = sa
Parametric Equations of Lines
x2
a
x1
Line passing through the origin:
x = sa
Parametric Equations of Lines
x2
a
x1
Line passing through the origin:
x = sa
Question: is this the only such equation for the line?
Parametric Equations of Lines
x2
a
x1
Line passing through the origin:
x = sa
Can we use this equation with a line not passing through the origin?
Parametric Equations of Lines
x2
a
q
x1
Parametric Equations of Lines
x2
a
q
x1
Parametric Equations of Lines
x2
a
q
x1
General equation of a line:
x = q + sa
x2
a
(1, 1)
x1
Finda parametric
equation describing the line in the direction of
−1
a=
, passing through the point (1, 1).
2
x2
a
(1, 1)
x1
Finda parametric
equation describing the line in the direction of
−1
a=
, passing through the point (1, 1).
2
x2
a
(1, 1)
x1
Finda parametric
equation describing the line in the direction of
−1
a=
, passing through the point (1, 1).
2
x2
a
(1, 1)
q
x1
Finda parametric
equation describing the line in the direction of
−1
a=
, passing through the point (1, 1).
2
x2
a
(1, 1)
q
x1
Finda parametric
equation describing the line in the direction of
−1
a=
, passing through the point (1, 1).
2
Can you find another?
x2
a
(1, 1)
q
x1
Finda parametric
equation describing the line in the direction of
−1
a=
, passing through the point (1, 1).
2
Can you find another?
Equations of Lines
x2
a
x1
Equations of Lines
x2
b
a
x1
Equations of Lines
x2
b
a
x1
Line passing through the origin:
Equations of Lines
x2
b
a
x1
Line passing through the origin:
b1
x
· 1 =0
b2
x2
Equations of Lines
x2
b
a
x1
Line passing through the origin:
b1
x
· 1 =0
b2
x2
⇒ b1 x1 + b2 x2 = 0
Equations of Lines
x2
b
a
x1
Line passing through the origin:
b1
x
· 1 =0
b2
x2
⇒ b1 x1 + b2 x2 = 0
⇒x2 = (−b2 /b1 )x1
Equations of Lines
x2
b
x1
General Line in R2
Equations of Lines
x2
b
x
q
x1
General Line in R2
Equations of Lines
x2
b
x
q
x−q
x1
General Line in R2
Equations of Lines
x2
b
x
q
x−q
x1
General Line in R2
(x − q) · b = 0
Equations of Lines
x2
b
x
q
x−q
x1
General Line in R2
(x − q) · b = 0
x·b=q·b
Equations of Lines
x2
b
x
q
x−q
x1
General Line in R2
(x − q) · b = 0
x·b=q·b
b1 x1 + b2 x2 = c
Suppose
theparametric
equation of a line is given by
x1
3
1
=
+s
. Convert this to an equation of the form
x2
−1
1
ax1 + bx2 = c.
Suppose
theparametric
equation of a line is given by
x1
3
2
=
+s
. Convert this to an equation of the form
x2
−1
7
ax1 + bx2 = c.
Suppose
theparametric
equation of a line is given by
x1
3
1
=
+s
. Convert this to an equation of the form
x2
−1
1
ax1 + bx2 = c.
x1
3
1
=
+s
x2
−1
1
x1 = 3 + s
⇔
x2 = −1 + s
Suppose
theparametric
equation of a line is given by
x1
3
1
=
+s
. Convert this to an equation of the form
x2
−1
1
ax1 + bx2 = c.
x1
3
1
=
+s
x2
−1
1
x1 = 3 + s
⇔
x2 = −1 + s
s = x1 − 3
⇔
s = x2 + 1
Suppose
theparametric
equation of a line is given by
x1
3
1
=
+s
. Convert this to an equation of the form
x2
−1
1
ax1 + bx2 = c.
x1
3
1
=
+s
x2
−1
1
x1 = 3 + s
⇔
x2 = −1 + s
s = x1 − 3
⇔
s = x2 + 1
⇒x1 − 3 = x2 + 1
⇔x1 − x2 = 2
Suppose
theparametric
equation of a line is given by
x1
3
1
=
+s
. Convert this to an equation of the form
x2
−1
1
ax1 + bx2 = c.
x1
3
1
=
+s
x2
−1
1
x1 = 3 + s
⇔
x2 = −1 + s
s = x1 − 3
⇔
s = x2 + 1
⇒x1 − 3 = x2 + 1
⇔x1 − x2 = 2
Suppose
theparametric
equation of a line is given by
x1
3
2
=
+s
. Convert this to an equation of the form
x2
−1
7
ax1 + bx2 = c.
Suppose
theparametric
equation of a line is given by
x1
3
1
=
+s
. Convert this to an equation of the form
x2
−1
1
ax1 + bx2 = c.
Suppose
theparametric
equation of a line is given by
x1
3
2
=
+s
. Convert this to an equation of the form
x2
−1
7
ax1 + bx2 = c.
x1 = 3 + 2s
x2 = −1 + 7s
x1 − 3 = 2s
⇔
x2 + 1 = 7s
7x1 − 21 = 14s
⇔
2x2 + 2 = 14s
⇒ 7x1 − 12 = 2x2 + 2
⇔ 7x1 − 2x2 = 23
Give a parametric equation for the line x2 = 3x1 + 5.
Give a parametric equation for the line x2 = 3x1 + 5.
a
Give a parametric equation for the line x2 = 3x1 + 5.
a
Give a parametric equation for the line x2 = 3x1 + 5.
q
a
Give a parametric equation for the line x2 = 3x1 + 5.
sa
q
a
q
a
q
a
x = q + sa
q
a
x = q + sa
How many components do the vectors have?
q
a
x = q + sa
How many dimensions does a vector have?
q
a
q
a
c
q
b
a
c
q
b
a
(x − q) · b = 0 and (x − q) · c = 0
Equation of a Line in R3
(x − q) · b = 0 and (x − q) · c = 0
Equation of a Line in R3
(x − q) · b = 0 and (x − q) · c = 0
x · b = q · b and x · c = q · c
Equation of a Line in R3
(x − q) · b = 0 and (x − q) · c = 0
x · b = q · b and x · c = q · c
To define a line in R3 , we need a system of equations:
x1 b1 + x2 b2 + x3 b3 = s1
x1 c1 + x2 c2 + x3 c3 = s2
q
a2
q
a1
a2
x
q
a1
a2
x
q
a1
x = q + sa1 + ta2
a2
q
a2
b
a2
q
a2
b
a2
x
q
a2
b
a2
x
q
a2
(x − q) · b = 0
b
a2
x
q
a2
(x − q) · b = 0
x·b=s
b
a2
x
q
a2
(x − q) · b = 0
x·b=s
b1 x1 + b2 x2 + b3 x3 = s
Equations
Line in R2
Line in
R3
Plane in R3
Parametric
x = q + sa
Component
b1 x1 + b2 x2 = s
x = q + sa
x = q + sa + tb
b1 x1 + b2 x2 + b3 x3 = s
c1 x1 + c2 x2 + c3 x3 = t
b1 x1 + b2 x2 + b3 x3 = s
Equations
Line in R2
Line in
R3
Plane in R3
Parametric
x = q + sa
Component
b1 x1 + b2 x2 = s
x = q + sa
x = q + sa + tb
b1 x1 + b2 x2 + b3 x3 = s
c1 x1 + c2 x2 + c3 x3 = t
b1 x1 + b2 x2 + b3 x3 = s
Suppose q and a are vectors in R18 . What would you call the geometric
object resulting from the equation x = q + sa?
Equations
Line in R2
Line in
R3
Plane in R3
Parametric
x = q + sa
Component
b1 x1 + b2 x2 = s
x = q + sa
x = q + sa + tb
b1 x1 + b2 x2 + b3 x3 = s
c1 x1 + c2 x2 + c3 x3 = t
b1 x1 + b2 x2 + b3 x3 = s
Suppose q and a are vectors in R18 . What would you call the geometric
object resulting from the equation x = q + sa?
Suppose P and Q are planes. What is the intersection of P and Q?
Equations
Line in R2
Line in
R3
Plane in R3
Parametric
x = q + sa
Component
b1 x1 + b2 x2 = s
x = q + sa
x = q + sa + tb
b1 x1 + b2 x2 + b3 x3 = s
c1 x1 + c2 x2 + c3 x3 = t
b1 x1 + b2 x2 + b3 x3 = s
Suppose q and a are vectors in R18 . What would you call the geometric
object resulting from the equation x = q + sa?
Suppose P and Q are planes. What is the intersection of P and Q?
Are there any vectors q and a in R3 for which the equation x = q + sa is
not a line?
Equations
Line in R2
Line in
R3
Plane in R3
Parametric
x = q + sa
Component
b1 x1 + b2 x2 = s
x = q + sa
x = q + sa + tb
b1 x1 + b2 x2 + b3 x3 = s
c1 x1 + c2 x2 + c3 x3 = t
b1 x1 + b2 x2 + b3 x3 = s
Suppose q and a are vectors in R18 . What would you call the geometric
object resulting from the equation x = q + sa?
Suppose P and Q are planes. What is the intersection of P and Q?
Are there any vectors q and a in R3 for which the equation x = q + sa is
not a line?
Are there any vectors q, a, and b in R3 for which the equation
x = q + sa + tb is not a plane?
Equations
Line in R2
Line in
R3
Plane in R3
Parametric
x = q + sa
Component
b1 x1 + b2 x2 = s
x = q + sa
x = q + sa + tb
b1 x1 + b2 x2 + b3 x3 = s
c1 x1 + c2 x2 + c3 x3 = t
b1 x1 + b2 x2 + b3 x3 = s
Are there any constants b1 , b2 , and s for which the equation
b1 x1 + b2 x2 = s is not a line?
Equations
Line in R2
Line in
R3
Plane in R3
Parametric
x = q + sa
Component
b1 x1 + b2 x2 = s
x = q + sa
x = q + sa + tb
b1 x1 + b2 x2 + b3 x3 = s
c1 x1 + c2 x2 + c3 x3 = t
b1 x1 + b2 x2 + b3 x3 = s
Are there any constants b1 , b2 , and s for which the equation
b1 x1 + b2 x2 = s is not a line?
Recall: b was the normal vector to the plane b1 x1 + b2 x2 + b3 x3 = s.
Equations
Line in R2
Line in
R3
Plane in R3
Parametric
x = q + sa
Component
b1 x1 + b2 x2 = s
x = q + sa
x = q + sa + tb
b1 x1 + b2 x2 + b3 x3 = s
c1 x1 + c2 x2 + c3 x3 = t
b1 x1 + b2 x2 + b3 x3 = s
Are there any constants b1 , b2 , and s for which the equation
b1 x1 + b2 x2 = s is not a line?
Recall: b was the normal vector to the plane b1 x1 + b2 x2 + b3 x3 = s.
True or False: for point P on the plane 5x1 + 7x2 + 11x3 = 22, the vector
with head at P and tail at the origin is orthogonal to the vector [5, 7, 11].
Equations
Line in R2
Line in
R3
Plane in R3
Parametric
x = q + sa
Component
b1 x1 + b2 x2 = s
x = q + sa
x = q + sa + tb
b1 x1 + b2 x2 + b3 x3 = s
c1 x1 + c2 x2 + c3 x3 = t
b1 x1 + b2 x2 + b3 x3 = s
Are there any constants b1 , b2 , and s for which the equation
b1 x1 + b2 x2 = s is not a line?
Recall: b was the normal vector to the plane b1 x1 + b2 x2 + b3 x3 = s.
True or False: for point P on the plane 5x1 + 7x2 + 11x3 = 22, the vector
with head at P and tail at the origin is orthogonal to the vector [5, 7, 11].
True or False: for any two distinct points P1 and P2 on the plane
5x1 + 7x2 + 11x3 = 22, the vector with head at P1 and tail at P2 is
orthogonal to the vector [5, 7, 11].
Prove It
Suppose a plane has equation b1 x1 + b2 x2 + b3 x3 = s.
Show that, for any two points on this plane, the vector with head at one
and tail at the other is orthogonal to [b1 , b2 , b3 ].
x2
b
x1
x2
x
b
x1
x2
x
b
projb x
x1
x2
x
b
projb x
x1
Draw lots of other vectors whose projections onto b are also the pink
vector.
x2
x
b
projb x
x1
Draw lots of other vectors whose projections onto b are also the pink
vector.
Find a simplified expression for the collection of vectors [x1 , x2 ] that all
have the same projection onto b in.
x2
a
P
q
x1
How can you find the distance from the point P to the line x = q + sa?
x2
a
d
P
q
x1
How can you find the distance from the point P to the line x = q + sa?
x2
a
d
P
q
x1
How can you find the distance from the point P to the line x = q + sa?
x2
d
a
P
q
v
x1
How can you find the distance from the point P to the line x = q + sa?
x2
proja v
d
a
P
q
v
x1
How can you find the distance from the point P to the line x = q + sa?
x2
proja v
d
a
P
q
v
x1
How can you find the distance from the point P to the line x = q + sa?
d + proja v =
x2
proja v
d
a
P
q
v
x1
How can you find the distance from the point P to the line x = q + sa?
d + proja v = v, so
x2
proja v
d
a
P
q
v
x1
How can you find the distance from the point P to the line x = q + sa?
d + proja v = v, so kdk = kv − proja vk
Find the distance from the point P to the plane Q.
Find the distance from the point P to the plane Q.
P
Find the distance from the point P to the plane Q.
P
Find the distance from the point P to the plane Q.
P
n
Find the distance from the point P to the plane Q.
P
v
n
Find the distance from the point P to the plane Q.
P
v
n
Find the distance from the point P to the plane Q.
P
projn v
v
n
Find the distance from the point P to the plane Q.
P
projn v
v
n
Findthe distance
  fromthepoint (3, 5, 1) to the plane
2
1
1
x = 6 + t 3 + s  1 .
3
5
−1
Let P be the plane with equation 2x + 2y + 2z = 1, and let Q be the
plane with equation x + y + z = 1.
What will their intersection be: a plane, a line, a point, or nothing?
Let P be the plane with equation 2x + 2y + 2z = 1, and let Q be the
plane with equation x + y + z = 1.
What will their intersection be: a plane, a line, a point, or nothing?
Let P be the plane with equation 2x + y − z = 1, and let Q be the plane
with equation x + 2y + 3z = 0.
What will their intersection be: a plane, a line, a point, or nothing?
Let P be the plane with equation 2x + 2y + 2z = 1, and let Q be the
plane with equation x + y + z = 1.
What will their intersection be: a plane, a line, a point, or nothing?
Let P be the plane with equation 2x + y − z = 1, and let Q be the plane
with equation x + 2y + 3z = 0.
What will their intersection be: a plane, a line, a point, or nothing?
Find it in parametric form.