vectors, dot product, planes, sections 12.2, 12nirobles/files241/lecture02.pdf · geometry of dot...
TRANSCRIPT
Math 241: Multivariable calculus, Lecture 2Vectors, Dot Product, Planes,
Sections 12.2, 12.3
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Wednesday, August 30th, 2017
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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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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.