[PDF] Dot product and vector projections (Sect 123) Two main ways to PDF L03-234.pdf

Dot product and orthogonal projections The dot product of the vectors v and w in Rn, with n = 2,3, Solution: The vector projection of b onto a is the vector p a

[PDF] 8-3 Dot Products and Vector Projections

Then write u as the sum of two orthogonal vectors, one of which is the projection of u onto v SOLUTION: Write u and v in component form as Find the projection

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b = a; by + azbe Notice that unlike vector addition and scalar multiplication, the dot product of two vectors yields a scalar, not a vector As demonstrated above, two

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Dot Products and Vector Projections Dot Product The dot product Use the dot product to find the magnitude of the given vector 3 a = 〈9, 3〉 4 c = 〈–12, 4〉

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the dot product of two-dimensional vectors is defined in a similar fashion: 〈a is π/3, find a b Solution: Using Theorem 3, we have a b = a b cos(π/3) = 4 6 = 12 magnitude of the vector projection, which is the number b cos θ

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Be able to use the dot product to find the angle between two vectors; and, the orthogonal projection of one vector onto another answer with HW 11 1 #3c )

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In Example 1, be sure you see that the dot product of two vectors is a scalar two orthogonal vectors, one of which is Solution The projection of onto is

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Two common operations involving vectors are the dot product and the cross product Let two vectors = , Solution: Using the first method of calculation, we have

pdf Dot product and vector projections (Sect 123) There are two

O Initial points together The dot product of two vectors is a scalar Example Compute v · w knowing that v w ? R3 with v = 2 w = h1 2 3i and the angle in between is ? = ?/4 Solution: We first compute w that is w2 = 12 + 22 + 32 = 14 ? ? w = 14 We now use the definition of dot product: ? ? 2 · w = v w cos(?) = (2) 14 2

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Searches related to 8 3 dot products and vector projections answers

points The dot product is also called scalar product or inner product It could be generalized Any product g(v;w) which is linear in vand wand satis es the symmetry g(v;w) = g(w;v) and g(v;v) 0 and g(v;v) = 0 if and only if v= 0 can be used as a dot product An example is g(v;w) = 2v 1w 1 + 3v 2w 2 + 5v 3w 3 2 8

Dot product and vector projections (Sect. 12.3)

Geometrical

definition→Properties→

Expression in

components.Definition in components→Properties→

Geometrical

expression.We choose the first way, the textbook chooses the second way.

Dot product and vector projections (Sect. 12.3)

Definition

Thedot productof the vectorsvandwinRn, withn= 2,3, having magnitudes|v|,|w|and angle in betweenθ, where v·w=|v||w|cos(θ).OV

WInitial points together.

The dot product of two vectors is a scalar

Example

Computev·wknowing thatv,w?R3, with|v|= 2,w=?1,2,3? and the angle in between isθ=π/4.Solution:We first compute|w|, that is, |w|2= 12+ 22+ 32= 14? |w|=⎷14.We now use the definition of dot product: v·w=|v||w|cos(θ) = (2)⎷14 ⎷2 2 ?v·w= 2⎷7.? ?The angle between two vectors usually is not know in applications.?It is useful to have a formula for the dot product involving the vector components.Dot product and vector projections (Sect. 12.3) ?Two definitions for the dot product. ?Geometric definition of dot product. ?Orthogonal vectors. ?Dot product and orthogonal projections. ?Properties of the dot product. ?Dot product in vector components. ?Scalar and vector projection formulas.

Perpendicular vectors have zero dot product.

Definition

Two vectors areperpendicular, also calledorthogonal, iff the angle in between isθ=π/2.0 = / 2

VWTheorem

The non-zero vectorsvandware perpendicular iffv·w= 0.Proof.

0 =v·w=|v||w|cos(θ)|v| ?= 0,|w| ?= 0?

?cos(θ) = 00?θ?π?θ=π2 .The dot product ofi,jandkis simple to computeExample

Compute all dot products involving the vectorsi,j, andk.Solution:Recall:i=?1,0,0?,j=?0,1,0?,k=?0,0,1?.

x ijk z yi·i= 1,j·j= 1,k·k= 1, i·j= 0,j·i= 0,k·i= 0, i·k= 0,j·k= 0,k·j= 0.?

Dot product and vector projections (Sect. 12.3)

?Two definitions for the dot product. ?Geometric definition of dot product. ?Orthogonal vectors. ?Dot product and orthogonal projections. ?Properties of the dot product. ?Dot product in vector components. ?Scalar and vector projection formulas.The dot product and orthogonal projections. Remark:The dot product is closely related to orthogonal projections of one vector onto the other.Recall:v·w=|v||w|cos(θ).WOV|v|cos(θ)= v·w|w|.WOV|w|cos(θ)= v·w|v|.Remark:If|u|= 1, thenv·uis the projection ofvalongu.

Dot product and vector projections (Sect. 12.3)

Theorem

(a)v·w=w·v,(symmetric); (b)v·(aw) =a(v·w),(linear); (c)u·(v+w) =u·v+u·w,(linear); (d)v·v=|v|2?0, andv·v= 0?v=0,(positive); (e)0·v= 0.Proof. Properties(a),(b),(d),(e)are simple to obtain from the

definition of dot productv·w=|v||w|cos(θ).For example, the proof of(b)fora>0:v·(aw) =|v||aw|cos(θ)=a|v||w|cos(θ)=a(v·w).

Properties of the dot product.

(c)u·(v+w) =u·v+u·w, is non-trivial.The proof is:W w V+W |V+W| cos( 0 ) V 0 0 W |V| cos( 0 )

V|W| cos( 0 )

W U0

V|v+w|cos(θ)=

u·(v+w)|u|,|w|cos(θw) =u·w|u|,|v|cos(θv) =u·v|u|,? ???????u·(v+w) =u·v+u·wDot product and vector projections (Sect. 12.3) ?Two definitions for the dot product. ?Geometric definition of dot product. ?Orthogonal vectors. ?Dot product and orthogonal projections. ?Properties of the dot product. ?Dot product in vector components. ?Scalar and vector projection formulas. The dot product in vector components (CaseR2)Theorem Ifv=?vx,vy?andw=?wx,wy?, thenv·wis given byv·w=vxwx+vywy.Proof. Recall:v=vxi+vyjandw=wxi+wyj.The linear property of the dot product implies

v·w= (vxi+vyj)·(wxi+wyj)v·w=vxwxi·i+vxwyi·j+vywxj·i+vywyj·j.Recall:i·i=j·j= 1 andi·j=j·i= 0.We conclude that

v·w=vxwx+vywy.The dot product in vector components (CaseR3)Theorem

Ifv=?vx,vy,vz?andw=?wx,wy,wz?, thenv·wis given byv·w=vxwx+vywy+vzwz.?The proof is similar to the case inR2.?The dot product is simple to compute from the vector

component formulav·w=vxwx+vywy+vzwz. ?The geometrical meaning of the dot product is simple to see from the formulav·w=|v||w|cos(θ).

Example

Find the cosine of the angle betweenv=?1,2?andw=?2,1?Solution: v·w=|v||w|cos(θ)?cos(θ) =v·w|v||w|.Furthermore, v·w= (1)(2) + (2)(1)|v|=?1

2+ 22=⎷5,|w|=?2

2+ 12=⎷5,?

???cos(θ) =45 .?Dot product and vector projections (Sect. 12.3) ?Two definitions for the dot product. ?Geometric definition of dot product. ?Orthogonal vectors. ?Dot product and orthogonal projections. ?Properties of the dot product. ?Dot product in vector components. ?Scalar and vector projection formulas.

Scalar and vector projection formulas.

Theorem

The scalar projection ofvalongwis the number pw(v),p w(v) =v·w|w|.The vector projection ofvalongwis the vectorpw(v),p w(v) =?v·w|w|? w|w|.| W | O | W |w V W

P ( V ) = ( V W ) W

| W | O | W |w V W

P ( V ) = ( V W ) WExample

Find the scalar projection ofb=?-4,1?ontoa=?1,2?.Solution:The scalar projection ofbontoais the number

p a(b) =|b|cos(θ)= b·a|a|= (-4)(1) + (1)(2)⎷1

2+ 22.We therefore obtainp

a(b) =-2⎷5. x p (b)a bay

Example

Find the vector projection ofb=?-4,1?ontoa=?1,2?.Solution:The vector projection ofbontoais the vector

p a(b) =?b·a|a|? a|a|= -2⎷5

1⎷5

?1,2?.We therefore obtainp a(b) =-?25 ,45 ?.x p (b)a bayExample

Find the vector projection ofa=?1,2?ontob=?-4,1?.Solution:The vector projection ofaontobis the vector

p b(a) =?a·b|b|? b|b|= -2⎷17

1⎷17

?-4,1?.We therefore obtainp a(b) =?817 ,-217 x ba p (a)b yquotesdbs_dbs11.pdfusesText_17

[PDF] [PDF] Dot product and vector projections (Sect 123) Two main ways to

Dot product and vector projections (Sect. 12.3)

Geometrical

Expression in

Geometrical

Dot product and vector projections (Sect. 12.3)

Definition

WInitial points together.

The dot product of two vectors is a scalar

Example

Perpendicular vectors have zero dot product.

Definition

VWTheorem

0 =v·w=|v||w|cos(θ)|v| ?= 0,|w| ?= 0?

Dot product and vector projections (Sect. 12.3)

Dot product and vector projections (Sect. 12.3)

Theorem

Properties of the dot product.

V|W| cos( 0 )

V|v+w|cos(θ)=

Example

2+ 22=⎷5,|w|=?2

2+ 12=⎷5,?

Scalar and vector projection formulas.

Theorem

P ( V ) = ( V W ) W

P ( V ) = ( V W ) WExample

2+ 22.We therefore obtainp

Example

1⎷5

1⎷17