**2.7.1. Vectorial Representation of Moments**. The moment of a force is a vector which is the product of distance and force. Hence in case of moment* of a force the cross-product of distance and force would be taken. Consider the Fig 2.31.

Let **F** = Force vector *(F _{x}*

*i*+

*F*j +

_{y}*F*

_{z}k)**r** = Distance (or position) vector with respect to O

*= **xi *+ *yj +zk *

*M *= Moment of force about point 0

* A quantity which is the product of two vectors and the quantity is also a vector, then cross product of the two vectors will be taken. But if the quantity is scalar, then dot product is taken.

*M _{z} = *Moment of F about z-axis.

Also *M _{x,} M_{y}, *and

*M*are known as scalar components of moment.

_{z}

**Problem 2.16.** *A **force *F=*2i *+ 4*j* – *3k **is applied at a point **P(*1*,*1, – 2). *Find the moment of the force F about the point *(2,-1, 2).

**Sol.**Given:

Force **F** = *2i *+ 4*j* – *3k *

The position vector **r** of the point *P *w.r.t. O. ,

= Position vector of point P

– Position vector of point O.

= *(i *+ *j* – *2k) *– *(2i *– j + *2k)*

**r** = (1- *2)i *+ [1 + (1)]*j* + [- 2 – 2]*k *

= -i + 2j – 4k

The moment M is given by

= [(2)(- 3) – (- 4)(4)] *i *+ [(- 4)(2) – (- 1) (- 3)]j + [(- 1)(4) – (2)(2)]*k
*=(- 6 + 16)

*i*+ (- 8 –

*3)j*+ (- 4 – 4)

*k*=

**10i -11j – 8k. Ans.**

**2.7.2. Vectorial Representation of Couples.** The moment produced by two equal, opposite parallel forces is known as couple.

Fig. 2.33. shows two equal opposite and parallel forces acting at points *A *and B. Let r_{A} and r_{B} are the position vectors of *A *and *B *with respect to O. The vector which joints *B to A *is represented by r.

The moment oftwo forces about point O is given by

Mo = rA × F- rB × F

= (rA – rB)× F

= r × F ( rA – rB=r)

The above equation shows that moment vector is independent of moment centre O.

M = r × F

This moment is known as couple.

The effect of couple is to produce pure rotation about an axis normal of the plane of force which constitute couple.

**2.7.3. Scaler Components of a Moment.** The moment of a force about any point O, is given by,

**M = r ****×**** F**

Where M* _{x}* = Moment of F about

*x*-axis = (

*y*F

*–*

_{z}*z*F

*),*

_{y}M* _{y}* = Moment of F about

*y*-axis = (

*z*F

*–*

_{x}*x*F

*) and*

_{z}M* _{z}* = Moment of F about

*z*-axis = (

*x*F

*–*

_{y}*y*F

*).*

_{x}Also M* _{x}*, M

*and M*

_{y}*are known as scaler components of Moment M.*

_{z}
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