Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN: 9780190698614
Author: Sadiku, Matthew N. O.
Publisher: Oxford University Press
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Chapter MA, Problem 1.8MA
To determine

To express: The unit vector an makes angles α,β and γ with the coordinate x-, y-, and z-axes, respectively. Express a non-unit vector OP of length l that is parallel to an.

Expert Solution & Answer
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Answer to Problem 1.8MA

The unit vector an makes angles α,β and γ with the coordinate x-, y-, and z-axes, respectively is expressed as cosαax+cosβay+cosγaz_ and a non-unit vector OP of length l that is parallel to an is expressed as lcosαax+lcosβay+lcosγaz_.

Explanation of Solution

Suppose the unit vector is an=xax+yay+zaz.

Perform dot product with ax on both sides of an=xax+yay+zaz.

anax=[xax+yay+zaz]ax|an||ax|cosα=x                                                  [axax=1,axay=0axaz=0]x=cosα                                                             [an and ax are unit vectors]

Perform dot product with ay on both sides of an=xax+yay+zaz.

anay=[xax+yay+zaz]ay|an||ay|cosβ=y                                                  [axax=1,axay=0axaz=0]y=cosβ                                                             [an and ay are unit vectors]

Perform dot product with az on both sides of an=xax+yay+zaz.

anaz=[xax+yay+zaz]az|an||az|cosγ=z                                                  [axax=1,axay=0axaz=0]z=cosγ                                                             [an and az are unit vectors]

Thus, the unit vector an can be expressed as an=cosαax+cosβay+cosγaz.

Consider the non-unit vector OP as OP=pax+qay+raz.

Perform dot product with an on both sides of OP=pax+qay+raz.

OPan=[pax+qay+raz]an|OP||an|cos0°=[pax+qay+raz][cosαax+cosβay+cosγaz]l=pcosα+qcosβ+rcosγ                                    [axax=1,axay=0, axaz=0,an is a unit vector and |OP|=l]                                                                                 

Perform cross product with an on both sides of OP=pax+qay+raz.

OP×an=|axayazpqrcosαcosβcosγ||OP||an|sin0°an=ax[qcosγrcosβ]ay[pcosγrcosα]                              +az[pcosβqcosα]                          [an is a unit vector and |OP|=l]ax[qcosγrcosβ]ay[pcosγrcosα]+az[pcosβqcosα]=0                                                                                 

Then, the system of equations is obtained as

qcosγrcosβ=0           ...(1)pcosγrcosα=0           ...(2)pcosβqcosα=0          ...(3)

From the equation 1 and 2, obtain the values of r and equate them.

r=qcosγcosβ and r=pcosγcosαpcosγcosα=qcosγcosβ           (Divide both sides by cosγ)p=qcosαcosβ

Substitute the value of p in equation 2 and obtain the value of r.

(qcosαcosβ)cosγrcosα=0                    (p=qcosαcosβ)rcosα=qcosαcosγcosβ                            (Divide both sides by cosα)r=qcosγcosβ

Substitute the values of p and r in the equation l=pcosα+qcosβ+rcosγ and obtain the value of q as follows.

l=(qcosαcosβ)cosα+qcosβ+(qcosγcosβ)cosγlcosβ=qcos2α+qcos2β+qcos2γq(cos2α+cos2β+cos2γ)=lcosβq=lcosβ                                               (|an|=cos2α+cos2β+cos2γ=1)

Substitute back the value of q in p and r in order to obtain their values.

That is,

p=lcosβcosαcosβ=lcosαr=lcosβcosγcosβ=lcosγ

Thus, the non-unit vector OP as OP=pax+qay+raz becomes OP=lcosαax+lcosβay+lcosγaz.

Therefore, the unit vector an makes angles α,β and γ with the coordinate x-, y-, and z-axes, respectively is expressed as cosαax+cosβay+cosγaz_ and a non-unit vector OP of length l that is parallel to an is expressed as lcosαax+lcosβay+lcosγaz_.

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