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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.Two main ways to introduce the dot product

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)

?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 of two vectors is a scalar

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)

?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.Properties of the dot product.

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

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