Question

show that the scalar product of two vectors is equal to the sum of the product of their rectangular components

Answer 1

A vector can be multiplied by a scalar.  The components of the vector are multiplied by the scalar and the result is a scaled vector which in the same direction as the original vector if the scalar is positive, or in the opposite direction if the scalar is negative.

A vector can also be multiplied by another vector.  Two types of vector multiplications have been defined, the scalar product and the vector product.  



The scalar product or dot product A×B of two vectors A and B is not a vector, but a scalar quantity (a number with units).  In terms of the Cartesian components of the vectors A and B the scalar product is written as

A×B = AxBx + AyBy + AzBz.

Consider two arbitrary vectors A and B and choose the orientation of your Cartesian coordinate system such that A points into the x-direction and B lies in the x-y plane.  Then A = (Ax, 0, 0) and B = (Bx, By, 0) and 

A×B = AxBx.  



Since Ax=A and Bx= Bcosf we can also write 

A×B = ABcosf .

The scalar product of two vectors A and B is a scalar quantity equal to the product of the magnitudes of the two vectors and the cosine of the smallest angle between them.  The scalar product is commutative, 

A×B = B×A.

When we form the scalar product of two vectors, we multiply the parallel component of the two vectors.

Many physical quantities of interest are calculated by forming the scalar product of two vectors.

Examples:

The work W done on an object by a constant force is defined as W=F×d.  It is equal to the magnitude of the force, multiplied by the distance the object moves in the direction of the force. The power P=F×v is the rate at which wok is done at an object.  We find it by multiplying the force acting on the object by the component of the velocity of the object perpendicular to the force.

To find the components of any vector quantity A along some arbitrary direction defined by the spherical angles q and f, form the unit vector 

u = (ux, uy, uz) =(sinq cosf, r sinq sinf, r cosq)

and then evaluate A×u = Axux + Ayuy + Azuz.

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The vector product or "cross product" of two vectors A and B is a vector C, defined as C=A´B.

We can find the Cartesian components of C=A´B in terms of the components of A and B.

Cx=AyBz-AzBy
Cy=AzBx-AxBz
Cz=AxBy-AyBx 

Again, consider two arbitrary vectors A and B and choose the orientation of your Cartesian coordinate system such that A points into the x-direction and B lies in the x-y plane.  Then A = (Ax, 0, 0) and B = (Bx, By, 0) and 

Cx=0
Cy=0
Cz=AxBy.

The magnitude of C is 

C=Cz=AxBy.



Since Ax=A and By= Bsinf we can also write

C=ABsinf, 

where f is the smallest angle between the directions of the vectors Aand B.  C is perpendicular to both A and B, i.e. it is perpendicular to the plane that contains both A and B.  The direction of C can be found by inspecting its components or by using the right-hand rule.

Let the fingers of your right hand point in the direction of A.  Orient the palm of your hand so that, as you curl your fingers, you can sweep them over to point in the direction of B.  Your thumb points in the direction of C=A´B.



If A and B are parallel or anti-parallel to each other, then C=A´B=0, since sinf=0.  If A and B are perpendicular to each other, then sinf=1 and C has its maximum possible magnitude.

When we form the scalar product of two vectors, we multiply the perpendicular component of the two vectors.  The vector product is not commutative, 

A´B = -B´A.

Many physical quantities of interest are calculated by forming the vector product of two vectors.

Examples:

A torque t is the product of a lever arm and a force that is applied perpendicular to the lever arm.  It is the vector product of r and F, t=r´F.

The angular momentum L of the particle about a point is L=r´p, where ris the displacement vector of the particle from the point and p is its momentum.  Only the component of p perpendicular to r contributes to the angular momentum L.

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Look at this 3D visualization of the cross product.



Vector Formulas:

A×(B´C)=B×(C´A)=C×(A´B)

A´(B´C)=B(A×)=C(A×)  (bac-cab rule)

(A´B)×(B´C)=(A×)(B×)-(A×)(B×)