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Express cross and dot product of two vectors in Cartesian coordinate.

Answer» <html><body><p></p>Solution :Let `vecA and vecB` be the two vectors. <br/> `vecA=vecA_(x)hati+vecA_(s)<a href="https://interviewquestions.tuteehub.com/tag/hatj-2693584" style="font-weight:bold;" target="_blank" title="Click to know more about HATJ">HATJ</a>+vecA_(z)hatk, vecB=vecB_(x)hati+vecB_(y)hatj+vecB_(z)hatk` <br/> Cross product of `vecA and vecB`. <br/> `<a href="https://interviewquestions.tuteehub.com/tag/vecaxxvecb-3257247" style="font-weight:bold;" target="_blank" title="Click to know more about VECAXXVECB">VECAXXVECB</a>=(vecA_(x)hati+vecA_(y)hatj+vecA_(z)hatk)<a href="https://interviewquestions.tuteehub.com/tag/xx-747671" style="font-weight:bold;" target="_blank" title="Click to know more about XX">XX</a>(vecB_(x)hati+vecB_(y)hatj+vecB_(z)hatk)` <br/> `=vecA_(x)vecB_(x)hatixxhati+vecA_(x)vecB_(y)hatixxhatj+vecA_(x)vecB_(z)hatixxhatk` <br/> `+vecA_(y)vecB_(y)hatjxxhati+vecA_(y)vecB_(z)hatjxxhatj+vecA_(z)B_(z)atkxxhatk` <br/> `vecAxxvecB=vecA_(x)vecB_(y)(hatk)+A_(x)B_(z)(-hatj)+vecA_(y)vecB_(x)(-hatk)+vecA_(y)vecB_(y)(0)` <br/> `+vecA_(y)vecB_(z)(hati)+vecA_(z)vecB_(x)(hatj)+vecA_(z)vecB_(z)(0)` <br/> `(vecAxxvecB)=(vecA_(y)vecB_(z)-vecA_(z)vecB_(y))hati+(vecA_(z)vecB_(x)-vecA_(x)vecB_(z))hatj+(vecA_(x)vecB_(y)-vecA_(y)vecB_(x))hatk` <br/> `[because hatixx hati=hatjxxhatj=hatkxxhatk=0` <br/> `hatixxhatj=k, hatixxhatk=-hatj, hatjxxhati=-k,hatjxxhatk=hati, hatk xxhati=+hatjxxhatj=-hatk]` <br/> It can be in <a href="https://interviewquestions.tuteehub.com/tag/determinant-949880" style="font-weight:bold;" target="_blank" title="Click to know more about DETERMINANT">DETERMINANT</a> form as, <br/> `vecA xx vecB=|(hati,hatj,hatjk),(A_(x),A_(y),A_(z)),(B_(x),B_(y),B_(z))|` <br/> `=hati(A_(y)B_(z)-B_(y)A_(z))-hatj(A_(x)B_(z)-B_(y)A_(z))+hatk(A_(x)B_(y)-B_(x)A_(y))` <br/> <a href="https://interviewquestions.tuteehub.com/tag/dot-958484" style="font-weight:bold;" target="_blank" title="Click to know more about DOT">DOT</a> product of `vecA and vecB`. <br/> `vecA=vecA_(x)hati+vecA_(y)hatj+vecA_(z)hatk, vecB=vecB_(x)hati+vecB_(x)hatj_vecB_(z)hatk` <br/> `vecA.vecB=(vecA_(x)hati+vecA_(y)hatj+vecA_(z)hatk).(vecB_(x)hati+vecB_(y)hatj+vecB_(z)hatk)` <br/> `(vecA_(x)vecB_(x)(hati.hatj)+vecA_(x)vecB_(y)(hati.hatj)` <br/> `+vecA_(x)vecB_(z)(hati.hatk)+vecA_(y)vecB_(x)(hatj.hatk)+vecA_(y)vecB_(y) (hati.hatj)` <br/> `+vecA_(y)vecB_(z)(hatj.hatk)+vecA_(z)vecB_(x)(hatk.hati)+vecA_(z)vecB_(y)(hatk.hatj)` <br/> `+A_(z)B_(z)(hatk.hatk)` <br/> `vecA.vecB=vec(A_(x))vec(B_(x))(1)+vec(A_(x))vec(B_(y))(0)+vecA_(x)vec(B_(z))(0)+vec(A_(y))vec(B_(z))(0)+vec(A_(y))vec(B_(y))(1)` <br/> `+vec(A_(y))vec(B_(z))(0)+vec(A_(z))vec(B_(x))(0)+vec(A_(z))v_(y)(0)+A_(z)B_(z)(1)` <br/> `vecA.vecB=vecA_(x)vecB_(x)+vec(A_(y))vec(B_(y))+vecA_(z)vecB_(z)` <br/> `[because hati.hati=hatj.hatj=hatk.hatk=1` <br/> `hati.hatj=hati.hatk=0, hatj.hati=hatj.hatk=0, hatk.hati=hatk.hatj=0]`</body></html>


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