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Question Number 139992 by ajfour last updated on 02/May/21

Commented by ajfour last updated on 03/May/21

$$Letroleofnewcross$$$$multiplicationby\hat{k}$$$$betorotatethecomponent$$$$ofavector\mathrm{\perp }tozaxisby90°$$$$counterclockwisewise.$$$$\Rightarrow k\times k=k,j\times j=j,i\times i=i,$$$$i\times k=j,j\times k=-i$$$$i\times j=-k,k\times j=i$$$$j\times i=k,k\times i=-j$$$$Now$$$$\overline{v}=xi+yj+zk$$$$\overline{p}=ai+bj+ck$$$$\overline{v}\times \overline{p}=axi-bxk+cxj$$$$+ayk+byj-cyi$$$$-azj+bzi+czk$$$$\Rightarrow \overline{v}\times \overline{p}=(ax-cy+bz)i+$$$$+(cx+by-az)j+(-bx+ay+cz)k$$$$Andif\overline{v}=\overline{p}$$$$\Rightarrow \overline{v}\times \overline{v}=x^{2}i+y^{2}j+z^{2}k$$$$\mid \overline{v}\times \overline{v}\mid =\sqrt{x^4+y^4+z^4}$$$$\mid \overline{v}\mid =\sqrt{x^2+y^2+z^2}$$$$\mid \overline{v}\times \overline{p}{\mid }^2={(ax-cy+bz)}^2$$$$+{(cx+by-az)}^2$$$$+{(-bx+ay+cz)}^2$$$$=(a^2+b^2+c^2)(x^2+y^2+z^2)$$$$+2(-ac+bc-ab)xy$$$$+2(-bc-ab+ac)yz$$$$+2(ab-ac-bc)zx$$$$Andifz=0,c=0;then$$$$\overline{v}=xi+yj$$$$\overline{p}=ai+bj$$$$\overline{v}\times \overline{p}=axi+byj+(-bx+ay)k$$$$\mid \overline{v}\times \overline{p}{\mid }^2=a^2{x}^2+b^2{y}^2+b^2{x}^2+a^2{y}^2$$$$-2abxy$$$$=(a^2+b^2)(x^2+y^2)-2abxy$$$$\Rightarrow \mid \overline{v}\times \overline{p}{\mid }^2=\mid \overline{p}{\mid }^2\mid \overline{v}{\mid }^2-2p_x{p}_y{v}_x{v}_y$$$$Butif\overline{v}=i+j+k$$$$\overline{v}\times \overline{v}=i+j+k$$$$Againif\overline{v}=xi+yj+zk$$$$\left(\overline{v}\times \overline{v}\right)\times \overline{v}=(x^{2}i+y^{2}j+z^{2}k)\times (xi+yj+zk)$$$$=x^{3}i-x^{2}yk+x^{2}zj$$$$+y^{2}xk+y^{3}j-y^{2}zi$$$$-z^{2}xj+z^{2}yi+z^{3}k$$$$\left(\overline{v}\times \overline{v}\right)\times \overline{v}=(x^3-y^{2}z+z^{2}y)i$$$$+(y^3-z^{2}x+x^{2}z)j$$$$+(z^3-x^{2}y+y^{2}x)k$$$$\left(\overline{v}\times \overline{v}\right)\times \overline{v}=0\Rightarrow$$$$x^3=yz(y-z)$$$$y^3=zx(z-x)$$$$z^3=xy(x-y)$$$$\Rightarrow x^4=m(y-z)wherem=xyz$$$$\Rightarrow x^4+y^4+z^4=0$$$$\frac{d\left(\overline{v}\times \overline{v}\right)}{dt}=2(x\frac{dx}{dt}i+y\frac{dy}{dt}j+z\frac{dz}{dt}k)$$$$$$

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