Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot...
Transcript of Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot...
![Page 1: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/1.jpg)
Math 241: Multivariable calculus, Lecture 2Vectors, Dot Product, Planes,
Sections 12.2, 12.3
go.illinois.edu/math241fa17
Wednesday, August 30th, 2017
go.illinois.edu/math241fa17.
![Page 2: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/2.jpg)
Math 241: Problems of the day
1. What is the equation of a sphere of radius 3 centered at(−1, 1, 0)?
2. What is the displacement vector ~v from the point (1, 2, 3) tothe point (3, 2, 1)? What is ‖~v‖? What does ‖~v‖ representgeometrically (with respect to the two points)?
go.illinois.edu/math241fa17.
![Page 3: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/3.jpg)
Math 241: Problems of the day
1. What is the equation of a sphere of radius 3 centered at(−1, 1, 0)?
2. What is the displacement vector ~v from the point (1, 2, 3) tothe point (3, 2, 1)? What is ‖~v‖? What does ‖~v‖ representgeometrically (with respect to the two points)?
go.illinois.edu/math241fa17.
![Page 4: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/4.jpg)
Last time: n-dimensional space and vectors.
Rn = {(x1, x2, . . . , xn) | xi ∈ R}
Distance between (a1, . . . , an) and (b1, . . . , bn) is√(b1 − a1)2 + . . .+ (bn − an)2.
Vectors are arrows, can identify them with n–tuples
Vectors in Rn ←→ Rn
−→OP ←→ P
〈v1, v2, . . . , vn〉 ←→ (v1, v2, dots, vn)
~v = 〈v1, . . . , vn〉 ⇒ v1, . . . , vn are components or coordinates.Displacement vector from A(a1, . . . , an) to B(b1, . . . , bn) is
−→AB = 〈b1 − a1, . . . , bn − an〉.
go.illinois.edu/math241fa17.
![Page 5: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/5.jpg)
Last time: n-dimensional space and vectors.
Rn = {(x1, x2, . . . , xn) | xi ∈ R}Distance between (a1, . . . , an) and (b1, . . . , bn) is√
(b1 − a1)2 + . . .+ (bn − an)2.
Vectors are arrows, can identify them with n–tuples
Vectors in Rn ←→ Rn
−→OP ←→ P
〈v1, v2, . . . , vn〉 ←→ (v1, v2, dots, vn)
~v = 〈v1, . . . , vn〉 ⇒ v1, . . . , vn are components or coordinates.Displacement vector from A(a1, . . . , an) to B(b1, . . . , bn) is
−→AB = 〈b1 − a1, . . . , bn − an〉.
go.illinois.edu/math241fa17.
![Page 6: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/6.jpg)
Last time: n-dimensional space and vectors.
Rn = {(x1, x2, . . . , xn) | xi ∈ R}Distance between (a1, . . . , an) and (b1, . . . , bn) is√
(b1 − a1)2 + . . .+ (bn − an)2.
Vectors are arrows, can identify them with n–tuples
Vectors in Rn ←→ Rn
−→OP ←→ P
〈v1, v2, . . . , vn〉 ←→ (v1, v2, dots, vn)
~v = 〈v1, . . . , vn〉 ⇒ v1, . . . , vn are components or coordinates.Displacement vector from A(a1, . . . , an) to B(b1, . . . , bn) is
−→AB = 〈b1 − a1, . . . , bn − an〉.
go.illinois.edu/math241fa17.
![Page 7: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/7.jpg)
Last time: n-dimensional space and vectors.
Rn = {(x1, x2, . . . , xn) | xi ∈ R}Distance between (a1, . . . , an) and (b1, . . . , bn) is√
(b1 − a1)2 + . . .+ (bn − an)2.
Vectors are arrows, can identify them with n–tuples
Vectors in Rn ←→ Rn
−→OP ←→ P
〈v1, v2, . . . , vn〉 ←→ (v1, v2, dots, vn)
~v = 〈v1, . . . , vn〉 ⇒ v1, . . . , vn are components or coordinates.
Displacement vector from A(a1, . . . , an) to B(b1, . . . , bn) is
−→AB = 〈b1 − a1, . . . , bn − an〉.
go.illinois.edu/math241fa17.
![Page 8: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/8.jpg)
Last time: n-dimensional space and vectors.
Rn = {(x1, x2, . . . , xn) | xi ∈ R}Distance between (a1, . . . , an) and (b1, . . . , bn) is√
(b1 − a1)2 + . . .+ (bn − an)2.
Vectors are arrows, can identify them with n–tuples
Vectors in Rn ←→ Rn
−→OP ←→ P
〈v1, v2, . . . , vn〉 ←→ (v1, v2, dots, vn)
~v = 〈v1, . . . , vn〉 ⇒ v1, . . . , vn are components or coordinates.Displacement vector from A(a1, . . . , an) to B(b1, . . . , bn) is
−→AB = 〈b1 − a1, . . . , bn − an〉.
go.illinois.edu/math241fa17.
![Page 9: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/9.jpg)
Addition, scalar multiplication, magnitude
Addition of vectors with “parallelogram rule” or componentwise:
〈u1, . . . , un〉+ 〈v1, . . . , vn〉 = 〈u1 + v1, . . . , un + vn〉.
Scalar multiplication: scale magnitude or componentwise:
c〈v1, . . . , vn〉 = 〈cv1, . . . , cvn〉.
Magnitude (or norm or length)
‖〈v1, . . . , vn〉‖ =√v21 + . . .+ v2n .
go.illinois.edu/math241fa17.
![Page 10: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/10.jpg)
Addition, scalar multiplication, magnitude
Addition of vectors with “parallelogram rule” or componentwise:
〈u1, . . . , un〉+ 〈v1, . . . , vn〉 = 〈u1 + v1, . . . , un + vn〉.
Scalar multiplication: scale magnitude or componentwise:
c〈v1, . . . , vn〉 = 〈cv1, . . . , cvn〉.
Magnitude (or norm or length)
‖〈v1, . . . , vn〉‖ =√v21 + . . .+ v2n .
go.illinois.edu/math241fa17.
![Page 11: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/11.jpg)
Addition, scalar multiplication, magnitude
Addition of vectors with “parallelogram rule” or componentwise:
〈u1, . . . , un〉+ 〈v1, . . . , vn〉 = 〈u1 + v1, . . . , un + vn〉.
Scalar multiplication: scale magnitude or componentwise:
c〈v1, . . . , vn〉 = 〈cv1, . . . , cvn〉.
Magnitude (or norm or length)
‖〈v1, . . . , vn〉‖ =√v21 + . . .+ v2n .
go.illinois.edu/math241fa17.
![Page 12: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/12.jpg)
Properties of vector arithmetic
~u, ~v , ~w vectors in Rn, and c , d ∈ R.
• ~u + (~v + ~w) = (~u + ~v) + ~w
• ~u +~0 = ~u
• ~u + (−~u) = ~0
• c(~u + ~v) = c~u + c~v
• (c + d)~u = c~u + d~u
• (cd)~u = c(d~u)
• 1~u = ~u.
go.illinois.edu/math241fa17.
![Page 13: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/13.jpg)
Properties of vector arithmetic
~u, ~v , ~w vectors in Rn, and c , d ∈ R.
• ~u + (~v + ~w) = (~u + ~v) + ~w
• ~u +~0 = ~u
• ~u + (−~u) = ~0
• c(~u + ~v) = c~u + c~v
• (c + d)~u = c~u + d~u
• (cd)~u = c(d~u)
• 1~u = ~u.
go.illinois.edu/math241fa17.
![Page 14: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/14.jpg)
Properties of vector arithmetic
~u, ~v , ~w vectors in Rn, and c , d ∈ R.
• ~u + (~v + ~w) = (~u + ~v) + ~w
• ~u +~0 = ~u
• ~u + (−~u) = ~0
• c(~u + ~v) = c~u + c~v
• (c + d)~u = c~u + d~u
• (cd)~u = c(d~u)
• 1~u = ~u.
go.illinois.edu/math241fa17.
![Page 15: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/15.jpg)
Properties of vector arithmetic
~u, ~v , ~w vectors in Rn, and c , d ∈ R.
• ~u + (~v + ~w) = (~u + ~v) + ~w
• ~u +~0 = ~u
• ~u + (−~u) = ~0
• c(~u + ~v) = c~u + c~v
• (c + d)~u = c~u + d~u
• (cd)~u = c(d~u)
• 1~u = ~u.
go.illinois.edu/math241fa17.
![Page 16: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/16.jpg)
Properties of vector arithmetic
~u, ~v , ~w vectors in Rn, and c , d ∈ R.
• ~u + (~v + ~w) = (~u + ~v) + ~w
• ~u +~0 = ~u
• ~u + (−~u) = ~0
• c(~u + ~v) = c~u + c~v
• (c + d)~u = c~u + d~u
• (cd)~u = c(d~u)
• 1~u = ~u.
go.illinois.edu/math241fa17.
![Page 17: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/17.jpg)
Properties of vector arithmetic
~u, ~v , ~w vectors in Rn, and c , d ∈ R.
• ~u + (~v + ~w) = (~u + ~v) + ~w
• ~u +~0 = ~u
• ~u + (−~u) = ~0
• c(~u + ~v) = c~u + c~v
• (c + d)~u = c~u + d~u
• (cd)~u = c(d~u)
• 1~u = ~u.
go.illinois.edu/math241fa17.
![Page 18: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/18.jpg)
Properties of vector arithmetic
~u, ~v , ~w vectors in Rn, and c , d ∈ R.
• ~u + (~v + ~w) = (~u + ~v) + ~w
• ~u +~0 = ~u
• ~u + (−~u) = ~0
• c(~u + ~v) = c~u + c~v
• (c + d)~u = c~u + d~u
• (cd)~u = c(d~u)
• 1~u = ~u.
go.illinois.edu/math241fa17.
![Page 19: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/19.jpg)
Standard basis vectors
In R2, define ~i = 〈1, 0〉, ~j = 〈0, 1〉. Every vector is a linearcombination of these:
〈v1, v2〉 = v1~i + v2~j
In R3 ⇒ ~i = 〈1, 0, 0〉, ~j = 〈0, 1, 0〉, ~k = 〈0, 0, 1〉
〈v1, v2, v3〉 = v1~i + v2~j + v3~k .
In Rn ⇒~e1 = 〈1, 0, . . . , 0〉, ~e2 = 〈0, 1, 0, . . . , 0〉, . . . , ~en = 〈0, . . . , 0, 1〉
〈v1, . . . , vn〉 = v1~e1 + . . .+ vn~en =n∑
j=1
vj~ej .
go.illinois.edu/math241fa17.
![Page 20: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/20.jpg)
Standard basis vectors
In R2, define ~i = 〈1, 0〉, ~j = 〈0, 1〉. Every vector is a linearcombination of these:
〈v1, v2〉 = v1~i + v2~j
In R3 ⇒ ~i = 〈1, 0, 0〉, ~j = 〈0, 1, 0〉, ~k = 〈0, 0, 1〉
〈v1, v2, v3〉 = v1~i + v2~j + v3~k .
In Rn ⇒~e1 = 〈1, 0, . . . , 0〉, ~e2 = 〈0, 1, 0, . . . , 0〉, . . . , ~en = 〈0, . . . , 0, 1〉
〈v1, . . . , vn〉 = v1~e1 + . . .+ vn~en =n∑
j=1
vj~ej .
go.illinois.edu/math241fa17.
![Page 21: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/21.jpg)
Standard basis vectors
In R2, define ~i = 〈1, 0〉, ~j = 〈0, 1〉. Every vector is a linearcombination of these:
〈v1, v2〉 = v1~i + v2~j
In R3 ⇒ ~i = 〈1, 0, 0〉, ~j = 〈0, 1, 0〉, ~k = 〈0, 0, 1〉
〈v1, v2, v3〉 = v1~i + v2~j + v3~k .
In Rn ⇒~e1 = 〈1, 0, . . . , 0〉, ~e2 = 〈0, 1, 0, . . . , 0〉, . . . , ~en = 〈0, . . . , 0, 1〉
〈v1, . . . , vn〉 = v1~e1 + . . .+ vn~en =n∑
j=1
vj~ej .
go.illinois.edu/math241fa17.
![Page 22: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/22.jpg)
Dot product
~u = 〈u1, . . . , un〉 and ~v = 〈v1, . . . , vn〉. The dot product of ~u and~v is the number
~u · ~v = u1v1 + . . .+ unvn =n∑
j=1
ujvj .
Easy properties: For vectors ~u, ~v , ~w and c ∈ R• ~u · (~v + ~w) = ~u · ~v + ~u · ~w ,
• ~u · ~v = ~v · ~u,
• (c~u) · ~v = c(~u · ~v) = ~v · (c~v),
• ~v · ~v = ‖~v‖2.
go.illinois.edu/math241fa17.
![Page 23: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/23.jpg)
Dot product
~u = 〈u1, . . . , un〉 and ~v = 〈v1, . . . , vn〉.
The dot product of ~u and~v is the number
~u · ~v = u1v1 + . . .+ unvn =n∑
j=1
ujvj .
Easy properties: For vectors ~u, ~v , ~w and c ∈ R• ~u · (~v + ~w) = ~u · ~v + ~u · ~w ,
• ~u · ~v = ~v · ~u,
• (c~u) · ~v = c(~u · ~v) = ~v · (c~v),
• ~v · ~v = ‖~v‖2.
go.illinois.edu/math241fa17.
![Page 24: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/24.jpg)
Dot product
~u = 〈u1, . . . , un〉 and ~v = 〈v1, . . . , vn〉. The dot product of ~u and~v is the number
~u · ~v = u1v1 + . . .+ unvn =n∑
j=1
ujvj .
Easy properties: For vectors ~u, ~v , ~w and c ∈ R• ~u · (~v + ~w) = ~u · ~v + ~u · ~w ,
• ~u · ~v = ~v · ~u,
• (c~u) · ~v = c(~u · ~v) = ~v · (c~v),
• ~v · ~v = ‖~v‖2.
go.illinois.edu/math241fa17.
![Page 25: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/25.jpg)
Dot product
~u = 〈u1, . . . , un〉 and ~v = 〈v1, . . . , vn〉. The dot product of ~u and~v is the number
~u · ~v = u1v1 + . . .+ unvn =n∑
j=1
ujvj .
Easy properties: For vectors ~u, ~v , ~w and c ∈ R
• ~u · (~v + ~w) = ~u · ~v + ~u · ~w ,
• ~u · ~v = ~v · ~u,
• (c~u) · ~v = c(~u · ~v) = ~v · (c~v),
• ~v · ~v = ‖~v‖2.
go.illinois.edu/math241fa17.
![Page 26: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/26.jpg)
Dot product
~u = 〈u1, . . . , un〉 and ~v = 〈v1, . . . , vn〉. The dot product of ~u and~v is the number
~u · ~v = u1v1 + . . .+ unvn =n∑
j=1
ujvj .
Easy properties: For vectors ~u, ~v , ~w and c ∈ R• ~u · (~v + ~w) = ~u · ~v + ~u · ~w ,
• ~u · ~v = ~v · ~u,
• (c~u) · ~v = c(~u · ~v) = ~v · (c~v),
• ~v · ~v = ‖~v‖2.
go.illinois.edu/math241fa17.
![Page 27: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/27.jpg)
Dot product
~u = 〈u1, . . . , un〉 and ~v = 〈v1, . . . , vn〉. The dot product of ~u and~v is the number
~u · ~v = u1v1 + . . .+ unvn =n∑
j=1
ujvj .
Easy properties: For vectors ~u, ~v , ~w and c ∈ R• ~u · (~v + ~w) = ~u · ~v + ~u · ~w ,
• ~u · ~v = ~v · ~u,
• (c~u) · ~v = c(~u · ~v) = ~v · (c~v),
• ~v · ~v = ‖~v‖2.
go.illinois.edu/math241fa17.
![Page 28: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/28.jpg)
Dot product
~u = 〈u1, . . . , un〉 and ~v = 〈v1, . . . , vn〉. The dot product of ~u and~v is the number
~u · ~v = u1v1 + . . .+ unvn =n∑
j=1
ujvj .
Easy properties: For vectors ~u, ~v , ~w and c ∈ R• ~u · (~v + ~w) = ~u · ~v + ~u · ~w ,
• ~u · ~v = ~v · ~u,
• (c~u) · ~v = c(~u · ~v) = ~v · (c~v),
• ~v · ~v = ‖~v‖2.
go.illinois.edu/math241fa17.
![Page 29: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/29.jpg)
Dot product
~u = 〈u1, . . . , un〉 and ~v = 〈v1, . . . , vn〉. The dot product of ~u and~v is the number
~u · ~v = u1v1 + . . .+ unvn =n∑
j=1
ujvj .
Easy properties: For vectors ~u, ~v , ~w and c ∈ R• ~u · (~v + ~w) = ~u · ~v + ~u · ~w ,
• ~u · ~v = ~v · ~u,
• (c~u) · ~v = c(~u · ~v) = ~v · (c~v),
• ~v · ~v = ‖~v‖2.
go.illinois.edu/math241fa17.
![Page 30: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/30.jpg)
Geometry of dot product
Key Theorem. Given vectors ~u and ~v in R2 or R3 making anangle 0 ≤ θ ≤ π, then
~u · ~v = ‖~u‖‖~v‖ cos(θ)
This comes from the law of cosines
|AB|2 = |AC |2 + |BC |2 − 2|AC ||BC | cos(θ)
Corollary. ~u ⊥ ~v if and only if ~u · ~v = 0.
~u ⊥ ~v means ~u and ~v are orthogonal (they make an θ = π2 or
90deg angle).
go.illinois.edu/math241fa17.
![Page 31: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/31.jpg)
Geometry of dot product
Key Theorem. Given vectors ~u and ~v in R2 or R3 making anangle 0 ≤ θ ≤ π, then
~u · ~v = ‖~u‖‖~v‖ cos(θ)
This comes from the law of cosines
|AB|2 = |AC |2 + |BC |2 − 2|AC ||BC | cos(θ)
Corollary. ~u ⊥ ~v if and only if ~u · ~v = 0.
~u ⊥ ~v means ~u and ~v are orthogonal (they make an θ = π2 or
90deg angle).
go.illinois.edu/math241fa17.
![Page 32: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/32.jpg)
Geometry of dot product
Key Theorem. Given vectors ~u and ~v in R2 or R3 making anangle 0 ≤ θ ≤ π, then
~u · ~v = ‖~u‖‖~v‖ cos(θ)
This comes from the law of cosines
|AB|2 = |AC |2 + |BC |2 − 2|AC ||BC | cos(θ)
Corollary. ~u ⊥ ~v if and only if ~u · ~v = 0.
~u ⊥ ~v means ~u and ~v are orthogonal (they make an θ = π2 or
90deg angle).
go.illinois.edu/math241fa17.
![Page 33: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/33.jpg)
Geometry of dot product
Key Theorem. Given vectors ~u and ~v in R2 or R3 making anangle 0 ≤ θ ≤ π, then
~u · ~v = ‖~u‖‖~v‖ cos(θ)
This comes from the law of cosines
|AB|2 = |AC |2 + |BC |2 − 2|AC ||BC | cos(θ)
Corollary. ~u ⊥ ~v if and only if ~u · ~v = 0.
~u ⊥ ~v means ~u and ~v are orthogonal (they make an θ = π2 or
90deg angle).
go.illinois.edu/math241fa17.
![Page 34: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/34.jpg)
Geometry of dot product
Key Theorem. Given vectors ~u and ~v in R2 or R3 making anangle 0 ≤ θ ≤ π, then
~u · ~v = ‖~u‖‖~v‖ cos(θ)
This comes from the law of cosines
|AB|2 = |AC |2 + |BC |2 − 2|AC ||BC | cos(θ)
Corollary. ~u ⊥ ~v if and only if ~u · ~v = 0.
~u ⊥ ~v means ~u and ~v are orthogonal (they make an θ = π2 or
90deg angle).
go.illinois.edu/math241fa17.
![Page 35: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/35.jpg)
Projections
The projection of ~v along ~u is the “part of ~v in the direction of ~u”:
proj~u~v =~u · ~v‖~u‖2
~u.
The basic property of projection is that ~v = proj~u~v + ~u⊥, with ~u⊥
perp to u. This leads to the formula.Example: What is proj~i 〈−2, 3, 7〉?
Answer: −2~i .
In general, proj~ej 〈v1, . . . , vn〉 = vj ~ej .
go.illinois.edu/math241fa17.
![Page 36: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/36.jpg)
Projections
The projection of ~v along ~u is the “part of ~v in the direction of ~u”:
proj~u~v =~u · ~v‖~u‖2
~u.
The basic property of projection is that ~v = proj~u~v + ~u⊥, with ~u⊥
perp to u. This leads to the formula.Example: What is proj~i 〈−2, 3, 7〉?
Answer: −2~i .
In general, proj~ej 〈v1, . . . , vn〉 = vj ~ej .
go.illinois.edu/math241fa17.
![Page 37: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/37.jpg)
Projections
The projection of ~v along ~u is the “part of ~v in the direction of ~u”:
proj~u~v =~u · ~v‖~u‖2
~u.
The basic property of projection is that ~v = proj~u~v + ~u⊥, with ~u⊥
perp to u. This leads to the formula.Example: What is proj~i 〈−2, 3, 7〉?
Answer: −2~i .
In general, proj~ej 〈v1, . . . , vn〉 = vj ~ej .
go.illinois.edu/math241fa17.
![Page 38: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/38.jpg)
Projections
The projection of ~v along ~u is the “part of ~v in the direction of ~u”:
proj~u~v =~u · ~v‖~u‖2
~u.
The basic property of projection is that ~v = proj~u~v + ~u⊥, with ~u⊥
perp to u. This leads to the formula.
Example: What is proj~i 〈−2, 3, 7〉?
Answer: −2~i .
In general, proj~ej 〈v1, . . . , vn〉 = vj ~ej .
go.illinois.edu/math241fa17.
![Page 39: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/39.jpg)
Projections
The projection of ~v along ~u is the “part of ~v in the direction of ~u”:
proj~u~v =~u · ~v‖~u‖2
~u.
The basic property of projection is that ~v = proj~u~v + ~u⊥, with ~u⊥
perp to u. This leads to the formula.Example: What is proj~i 〈−2, 3, 7〉?
Answer: −2~i .
In general, proj~ej 〈v1, . . . , vn〉 = vj ~ej .
go.illinois.edu/math241fa17.
![Page 40: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/40.jpg)
Projections
The projection of ~v along ~u is the “part of ~v in the direction of ~u”:
proj~u~v =~u · ~v‖~u‖2
~u.
The basic property of projection is that ~v = proj~u~v + ~u⊥, with ~u⊥
perp to u. This leads to the formula.Example: What is proj~i 〈−2, 3, 7〉?
Answer: −2~i .
In general, proj~ej 〈v1, . . . , vn〉 = vj ~ej .
go.illinois.edu/math241fa17.
![Page 41: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/41.jpg)
Projections
The projection of ~v along ~u is the “part of ~v in the direction of ~u”:
proj~u~v =~u · ~v‖~u‖2
~u.
The basic property of projection is that ~v = proj~u~v + ~u⊥, with ~u⊥
perp to u. This leads to the formula.Example: What is proj~i 〈−2, 3, 7〉?
Answer: −2~i .
In general, proj~ej 〈v1, . . . , vn〉 = vj ~ej .
go.illinois.edu/math241fa17.
![Page 42: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/42.jpg)
Application: Work
Work = force × distance
Force and Distance are vectors, Work is a number. So moreprecisely:
W = ‖proj~D ~F‖‖ ~D‖
=∥∥∥ ~F ·~D‖~D‖2
~D∥∥∥ ‖ ~D‖
= |~F · ~D|‖~D‖2
‖~D‖2
= ~F · ~D
B
A
D
F
go.illinois.edu/math241fa17.
![Page 43: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/43.jpg)
Application: Work
Work = force × distance
Force and Distance are vectors, Work is a number. So moreprecisely:
W = ‖proj~D ~F‖‖ ~D‖
=∥∥∥ ~F ·~D‖~D‖2
~D∥∥∥ ‖ ~D‖
= |~F · ~D|‖~D‖2
‖~D‖2
= ~F · ~D
B
A
D
F
go.illinois.edu/math241fa17.
![Page 44: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/44.jpg)
Application: Work
Work = force × distance
Force and Distance are vectors, Work is a number. So moreprecisely:
W = ‖proj~D ~F‖‖ ~D‖
=∥∥∥ ~F ·~D‖~D‖2
~D∥∥∥ ‖ ~D‖
= |~F · ~D|‖~D‖2
‖~D‖2
= ~F · ~D
B
A
D
F
go.illinois.edu/math241fa17.
![Page 45: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/45.jpg)
Application: Work
Work = force × distance
Force and Distance are vectors, Work is a number. So moreprecisely:
W = ‖proj~D ~F‖‖ ~D‖
=∥∥∥ ~F ·~D‖~D‖2
~D∥∥∥ ‖ ~D‖
= |~F · ~D|‖~D‖2
‖~D‖2
= ~F · ~D
B
A
D
F
go.illinois.edu/math241fa17.
![Page 46: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/46.jpg)
Application: Work
Work = force × distance
Force and Distance are vectors, Work is a number. So moreprecisely:
W = ‖proj~D ~F‖‖ ~D‖
=∥∥∥ ~F ·~D‖~D‖2
~D∥∥∥ ‖ ~D‖
= |~F · ~D|‖~D‖2
‖~D‖2
= ~F · ~D
B
A
D
F
go.illinois.edu/math241fa17.
![Page 47: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/47.jpg)
Application: Work
Work = force × distance
Force and Distance are vectors, Work is a number. So moreprecisely:
W = ‖proj~D ~F‖‖ ~D‖
=∥∥∥ ~F ·~D‖~D‖2
~D∥∥∥ ‖ ~D‖
= |~F · ~D|‖~D‖2
‖~D‖2
= ~F · ~D
B
A
D
F
go.illinois.edu/math241fa17.
![Page 48: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/48.jpg)
Application: Work
Work = force × distance
Force and Distance are vectors, Work is a number. So moreprecisely:
W = ‖proj~D ~F‖‖ ~D‖
=∥∥∥ ~F ·~D‖~D‖2
~D∥∥∥ ‖ ~D‖
= |~F · ~D|‖~D‖2
‖~D‖2
= ~F · ~D
B
A
D
F
go.illinois.edu/math241fa17.
![Page 49: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/49.jpg)
Application: Work
Work = force × distance
Force and Distance are vectors, Work is a number. So moreprecisely:
W = ‖proj~D ~F‖‖ ~D‖
=∥∥∥ ~F ·~D‖~D‖2
~D∥∥∥ ‖ ~D‖
= |~F · ~D|‖~D‖2
‖~D‖2
= ~F · ~D
B
A
D
F
go.illinois.edu/math241fa17.
![Page 50: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/50.jpg)
Application: Equation of a plane
P0(x0, y0, z0) point in the plane, ~n = 〈a, b, c〉 normal vector tothe plane (=vector orthogonal to the plane).Equation:
ax + by + cz = (ax0 + by0 + cz0).
Why?
Example. Find equation of the plane through the point (1,−1, 2)parallel to the plane x + y + z = 0.
n
(x ,y ,z )0 0 0
go.illinois.edu/math241fa17.
![Page 51: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/51.jpg)
Application: Equation of a plane
P0(x0, y0, z0) point in the plane,
~n = 〈a, b, c〉 normal vector tothe plane (=vector orthogonal to the plane).Equation:
ax + by + cz = (ax0 + by0 + cz0).
Why?
Example. Find equation of the plane through the point (1,−1, 2)parallel to the plane x + y + z = 0.
n
(x ,y ,z )0 0 0
go.illinois.edu/math241fa17.
![Page 52: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/52.jpg)
Application: Equation of a plane
P0(x0, y0, z0) point in the plane, ~n = 〈a, b, c〉 normal vector tothe plane (=vector orthogonal to the plane).
Equation:
ax + by + cz = (ax0 + by0 + cz0).
Why?
Example. Find equation of the plane through the point (1,−1, 2)parallel to the plane x + y + z = 0.
n
(x ,y ,z )0 0 0
go.illinois.edu/math241fa17.
![Page 53: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/53.jpg)
Application: Equation of a plane
P0(x0, y0, z0) point in the plane, ~n = 〈a, b, c〉 normal vector tothe plane (=vector orthogonal to the plane).Equation:
ax + by + cz = (ax0 + by0 + cz0).
Why?
Example. Find equation of the plane through the point (1,−1, 2)parallel to the plane x + y + z = 0.
n
(x ,y ,z )0 0 0
go.illinois.edu/math241fa17.
![Page 54: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/54.jpg)
Application: Equation of a plane
P0(x0, y0, z0) point in the plane, ~n = 〈a, b, c〉 normal vector tothe plane (=vector orthogonal to the plane).Equation:
ax + by + cz = (ax0 + by0 + cz0).
Why?
Example. Find equation of the plane through the point (1,−1, 2)parallel to the plane x + y + z = 0.
n
(x ,y ,z )0 0 0
go.illinois.edu/math241fa17.
![Page 55: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/55.jpg)
Application: Equation of a plane
P0(x0, y0, z0) point in the plane, ~n = 〈a, b, c〉 normal vector tothe plane (=vector orthogonal to the plane).Equation:
ax + by + cz = (ax0 + by0 + cz0).
Why?
Example. Find equation of the plane through the point (1,−1, 2)parallel to the plane x + y + z = 0.
n
(x ,y ,z )0 0 0
go.illinois.edu/math241fa17.
![Page 56: Vectors, Dot Product, Planes, Sections 12.2, 12nirobles/files241/lecture02.pdf · Geometry of dot product 2 making an angle 0 ˇ, then u~~v= k~ukk~vkcos( ) This comes from the law](https://reader033.fdocuments.us/reader033/viewer/2022053021/604a752e1f24e81ab33f5d86/html5/thumbnails/56.jpg)
Application: Equation of a plane
P0(x0, y0, z0) point in the plane, ~n = 〈a, b, c〉 normal vector tothe plane (=vector orthogonal to the plane).Equation:
ax + by + cz = (ax0 + by0 + cz0).
Why?
Example. Find equation of the plane through the point (1,−1, 2)parallel to the plane x + y + z = 0.
n
(x ,y ,z )0 0 0
go.illinois.edu/math241fa17.