Answer to Question #297478 in Differential Geometry | Topology for Imakp

Question #297478

Vector A=2ti+tj-t^3k and B=sinti+costj evaluate

A..d/dt(A.B)

B..d/dt(A.A)

C..d/dt(A×B)

D..show that d/dt(A×A) is equal to zero.


1
Expert's answer
2022-02-14T18:33:30-0500

A.


"A\\cdot B=2t\\sin t+t\\cos t"

"\\dfrac{d}{dt}(A\\cdot B)=2\\sin t+2t\\cos t+\\cos t-t\\sin t"

B.



"A\\cdot A=4t^2+t^2=5t^2"

"\\dfrac{d}{dt}(A\\cdot A)=10t"

C.


"A\\times B=\\begin{vmatrix}\n i & j & k \\\\\n 2t & t & -t^3 \\\\\n \\sin t & \\cos t & 0 \\\\\n\\end{vmatrix}"

"=i(0+t^3\\cos t)-j(0+t^3\\sin t)+k(2t\\cos t-t\\sin t)"

"\\dfrac{d}{dt}(A\\times B)=(3t^2\\cos t-t^3\\sin t)i-(3t^2\\sin t+t^3\\cos t)j"

"+(2\\cos t-2t\\sin t-\\sin t-t\\cos t)k"

D.


"A\\times A=\\begin{vmatrix}\n i & j & k \\\\\n 2t & t & -t^3 \\\\\n 2t & t & -t^3 \\\\\n\\end{vmatrix}"

"=i(-t^4+t^4)-j(-2t^4+2t^4)+k(2t^2-2t^2)=0"

"\\dfrac{d}{dt}(A\\times A)=0"



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