Question

dot product is commutative while cross product of vectors is not commutative so this with an example of vectors

Answer 1

Vectors and dot products¶

A vector is an ordered sequence of values:

v⃗ =[v1,v2,⋯vn]

Vector scaling

A vector can be scaled by a scalar cc:

cv⃗ ≜[cv1,cv2,⋯cvn]cv→≜[cv1,cv2,⋯cvn]

Vector addition

Say we have two vectors containing nn values:

v⃗ =[v1,v2,⋯vn]w⃗ =[w1,w2,⋯wn]v→=[v1,v2,⋯vn]w→=[w1,w2,⋯wn]

Vector addition gives a new vector with nn values:

v⃗ +w⃗ ≜[v1+w1,v2+w2,⋯vn+wn]v→+w→≜[v1+w1,v2+w2,⋯vn+wn]

Vector addition is commutative because vi+wi=wi+vivi+wi=wi+vi:

v⃗ +w⃗ =w⃗ +v⃗ v→+w→=w→+v→

Vector dot product

The vector dot product is:

v⃗ ⋅w⃗ ≜Σni=1viwiv→⋅w→≜Σi=1nviwi

Vector length

We write the length of a vector v⃗ v→as ∥v⃗ ∥‖v→‖:

∥v⃗ ∥≜Σv2i−−−√‖v→‖≜Σvi2

This is a generalization of Pythagoras’ theorem to nndimensions. For example, the length of a two dimensional vector [x,y][x,y] is the length of the hypotenuse of the right-angle triangle formed by the points (x,0),(0,y),(x,y)(x,0),(0,y),(x,y). This length is x2+y2−−−−−−√x2+y2. For a point in three dimensions x,y,zx,y,z, consider the right-angle triangle formed by (x,y,0),(0,0,z),(x,y,z)(x,y,0),(0,0,z),(x,y,z). The hypotenuse is length ∥[x,y]∥2+z2−−−−−−−−−−−√=x2+y2+z2−−−−−−−−−−√‖[x,y]‖2+z2=x2+y2+z2.

From the definition of vector length and the dot product, the square root of the dot product of the vector with itself gives the vector length:

∥v⃗ ∥=v⃗ ⋅v⃗ −−−√‖v→‖=v→⋅v→

Properties of dot products

We will use the results from Some algebra with summation.

Commutative

v⃗ ⋅w⃗ =w⃗ ⋅v⃗ v→⋅w→=w→⋅v→

because viwi=wiviviwi=wivi.

Distributive over vector addition

v⃗ ⋅(w⃗ +x⃗ )=v⃗ ⋅w⃗ +v⃗ ⋅x⃗ v→⋅(w→+x→)=v→⋅w→+v→⋅x→

because:

v⃗ ⋅(w⃗ +x⃗ )=Σvi(wi+xi)=Σ(vi+wi)+Σ(vi+xi)=v⃗ ⋅w⃗ +v⃗ ⋅x⃗ v→⋅(w→+x→)=Σvi(wi+xi)=Σ(vi+wi)+Σ(vi+xi)=v→⋅w→+v→⋅x→

Scalar multiplication

Say we have two scalars, cc and dd:

(cv⃗ )⋅(dw⃗ )=cd(v⃗ ⋅w⃗ )(cv→)⋅(dw→)=cd(v→⋅w→)

because:

(cv⃗ )⋅(dw⃗ )=Σcvidwi=cdΣviwi(cv→)⋅(dw→)=Σcvidwi=cdΣviwi

From the properties of distribution over addition and scalar multiplication:

v⃗ ⋅(cw⃗ +x⃗ )=c(v⃗ ⋅w⃗ )+(v⃗ ⋅x⃗ )v→⋅(cw→+x→)=c(v→⋅w→)+(v→⋅x→)

Unit vector

A unit vector is any vector with length 1.

To make a corresponding unit vector from any vector v⃗ v→, divide by ∥v⃗ ∥‖v→‖:

u⃗ =1∥v⃗ ∥v⃗ u→=1‖v→‖v→

Let g≜1∥v⃗ ∥g≜1‖v→‖. Then:

∥gv⃗ ∥2=(gv⃗ )⋅(gv⃗ )=g2∥v⃗ ∥2=1‖gv→‖2=(gv→)⋅(gv→)=g2‖v→‖2=1

If two vectors are perpendicular, their dot product is 0

I based this proof on that in Gilbert Strang’s “Introduction to Linear Algebra” 4th edition, page 14.

Consider the triangle formed by the two vectors v⃗ v→ and w⃗ w→. The lengths of the sides of the triangle are ∥v⃗ ∥,∥w⃗ ∥,∥v⃗ −w⃗ ∥‖v→‖,‖w→‖,‖v→−w→‖. When v⃗ v→ and w⃗ w→ are perpendicular, this is a right-angled triangle with hypotenuse length ∥v⃗ −w⃗ ∥‖v→−w→‖. In this situation, by Pythagoras:

∥v⃗ ∥2+∥w⃗ ∥2=∥v⃗ −w⃗ ∥2‖v→‖2+‖w→‖2=‖v→−w→‖2

Write the left hand side as:

∥v⃗ ∥2+∥w⃗ ∥2=v21+v22+⋯v2n+w21+w22+⋯w2n‖v→‖2+‖w→‖2=v12+v22+⋯vn2+w12+w22+⋯wn2

Write the right hand side as:

∥v⃗ −w⃗ ∥2=(v21−2v1w1+w21)+(v22−2v2w2+w22)+⋯(v2n−2vnw1+w2n)‖v→−w→‖2=(v12−2v1w1+w12)+(v22−2v2w2+w22)+⋯(vn2−2vnw1+wn2)

The v2ivi2 and w2iwi2 terms on left and right cancel, so:

∥v⃗ ∥2+∥w⃗ ∥2=∥v⃗ −w⃗ ∥2⟹0=2(v1w1+v2w2+⋯vnwn)⟹0=v⃗ ⋅w⃗ ‖v→‖2+‖w→‖2=‖v→−w→‖2⟹0=2(v1w1+v2w2+⋯vnwn)⟹0=v→⋅w→

By the converse of pythagoras , theorem if ∥v⃗ ∥2+∥w⃗ ∥2≠∥v⃗ −w⃗ ∥2‖v→‖2+‖w→‖2≠‖v→−w→‖2then vectors v⃗ v→ and w⃗ w→ do not form a right angle and are not perpendicular.

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