vectors, dot product, planes, sections 12.2, 12nirobles/files241/lecture02.pdf · geometry of dot...

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.

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)?

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)?

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〉.

Rn = {(x1, x2, . . . , xn) | xi ∈ R}Distance between (a1, . . . , an) and (b1, . . . , bn) is√

(b1 − a1)2 + . . .+ (bn − an)2.

−→OP ←→ P

〈v1, v2, . . . , vn〉 ←→ (v1, v2, dots, vn)

−→AB = 〈b1 − a1, . . . , bn − an〉.

(b1 − a1)2 + . . .+ (bn − an)2.

−→OP ←→ P

〈v1, v2, . . . , vn〉 ←→ (v1, v2, dots, vn)

−→AB = 〈b1 − a1, . . . , bn − an〉.

(b1 − a1)2 + . . .+ (bn − an)2.

−→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〉.

(b1 − a1)2 + . . .+ (bn − an)2.

−→OP ←→ P

〈v1, v2, . . . , vn〉 ←→ (v1, v2, dots, vn)

−→AB = 〈b1 − a1, . . . , bn − an〉.

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 .

〈u1, . . . , un〉+ 〈v1, . . . , vn〉 = 〈u1 + v1, . . . , un + vn〉.

c〈v1, . . . , vn〉 = 〈cv1, . . . , cvn〉.

‖〈v1, . . . , vn〉‖ =√v21 + . . .+ v2n .

〈u1, . . . , un〉+ 〈v1, . . . , vn〉 = 〈u1 + v1, . . . , un + vn〉.

c〈v1, . . . , vn〉 = 〈cv1, . . . , cvn〉.

‖〈v1, . . . , vn〉‖ =√v21 + . . .+ v2n .

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.

• ~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.

• ~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.

• ~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.

• ~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.

• ~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.

• ~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.

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∑

vj~ej .

〈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∑

vj~ej .

〈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∑

vj~ej .

Dot product

~u = 〈u1, . . . , un〉 and ~v = 〈v1, . . . , vn〉. The dot product of ~u and~v is the number

~u · ~v = u1v1 + . . .+ unvn =n∑

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.

Dot product

~u = 〈u1, . . . , un〉 and ~v = 〈v1, . . . , vn〉.

The dot product of ~u and~v is the number

~u · ~v = u1v1 + . . .+ unvn =n∑

ujvj .

• ~u · ~v = ~v · ~u,

• (c~u) · ~v = c(~u · ~v) = ~v · (c~v),

• ~v · ~v = ‖~v‖2.

Dot product

~u · ~v = u1v1 + . . .+ unvn =n∑

ujvj .

• ~u · ~v = ~v · ~u,

• (c~u) · ~v = c(~u · ~v) = ~v · (c~v),

• ~v · ~v = ‖~v‖2.

Dot product

~u · ~v = u1v1 + . . .+ unvn =n∑

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.

Dot product

~u · ~v = u1v1 + . . .+ unvn =n∑

ujvj .

• ~u · ~v = ~v · ~u,

• (c~u) · ~v = c(~u · ~v) = ~v · (c~v),

• ~v · ~v = ‖~v‖2.

Dot product

~u · ~v = u1v1 + . . .+ unvn =n∑

ujvj .

• ~u · ~v = ~v · ~u,

• (c~u) · ~v = c(~u · ~v) = ~v · (c~v),

• ~v · ~v = ‖~v‖2.

Dot product

~u · ~v = u1v1 + . . .+ unvn =n∑

ujvj .

• ~u · ~v = ~v · ~u,

• (c~u) · ~v = c(~u · ~v) = ~v · (c~v),

• ~v · ~v = ‖~v‖2.

Dot product

~u · ~v = u1v1 + . . .+ unvn =n∑

ujvj .

• ~u · ~v = ~v · ~u,

• (c~u) · ~v = c(~u · ~v) = ~v · (c~v),

• ~v · ~v = ‖~v‖2.

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).

~u · ~v = ‖~u‖‖~v‖ cos(θ)

|AB|2 = |AC |2 + |BC |2 − 2|AC ||BC | cos(θ)

90deg angle).

~u · ~v = ‖~u‖‖~v‖ cos(θ)

|AB|2 = |AC |2 + |BC |2 − 2|AC ||BC | cos(θ)

90deg angle).

~u · ~v = ‖~u‖‖~v‖ cos(θ)

|AB|2 = |AC |2 + |BC |2 − 2|AC ||BC | cos(θ)

90deg angle).

~u · ~v = ‖~u‖‖~v‖ cos(θ)

|AB|2 = |AC |2 + |BC |2 − 2|AC ||BC | cos(θ)

90deg angle).

Projections

The projection of ~v along ~u is the “part of ~v in the direction of ~u”:

proj~u~v =~u · ~v‖~u‖2

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 .

Projections

Answer: −2~i .

Projections

Answer: −2~i .

Projections

perp to u. This leads to the formula.

Example: What is proj~i 〈−2, 3, 7〉?

Answer: −2~i .

Projections

Answer: −2~i .

Projections

Answer: −2~i .

Projections

Answer: −2~i .

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

Application: Work

=∥∥∥ ~F ·~D‖~D‖2

~D∥∥∥ ‖ ~D‖

= |~F · ~D|‖~D‖2

‖~D‖2

= ~F · ~D

Application: Work

=∥∥∥ ~F ·~D‖~D‖2

~D∥∥∥ ‖ ~D‖

= |~F · ~D|‖~D‖2

‖~D‖2

= ~F · ~D

Application: Work

=∥∥∥ ~F ·~D‖~D‖2

~D∥∥∥ ‖ ~D‖

= |~F · ~D|‖~D‖2

‖~D‖2

= ~F · ~D

Application: Work

=∥∥∥ ~F ·~D‖~D‖2

~D∥∥∥ ‖ ~D‖

= |~F · ~D|‖~D‖2

‖~D‖2

= ~F · ~D

Application: Work

=∥∥∥ ~F ·~D‖~D‖2

~D∥∥∥ ‖ ~D‖

= |~F · ~D|‖~D‖2

‖~D‖2

= ~F · ~D

Application: Work

=∥∥∥ ~F ·~D‖~D‖2

~D∥∥∥ ‖ ~D‖

= |~F · ~D|‖~D‖2

‖~D‖2

= ~F · ~D

Application: Work

=∥∥∥ ~F ·~D‖~D‖2

~D∥∥∥ ‖ ~D‖

= |~F · ~D|‖~D‖2

‖~D‖2

= ~F · ~D

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).

Example. Find equation of the plane through the point (1,−1, 2)parallel to the plane x + y + z = 0.

(x ,y ,z )0 0 0

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).

(x ,y ,z )0 0 0

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).

(x ,y ,z )0 0 0

ax + by + cz = (ax0 + by0 + cz0).

(x ,y ,z )0 0 0

ax + by + cz = (ax0 + by0 + cz0).

(x ,y ,z )0 0 0

ax + by + cz = (ax0 + by0 + cz0).

(x ,y ,z )0 0 0

ax + by + cz = (ax0 + by0 + cz0).

(x ,y ,z )0 0 0

vectors, dot product, planes, sections 12.2, 12nirobles/files241/lecture02.pdf · geometry of dot...

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