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The systems to the vector is below:

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Given(R(t)=)and point()

The point()occursat t=-2

Find the derivative the the vector,

(R"(t)=)

(|R"(t)|=sqrt(2t)^2+(2t^2)^2+1^2)

(=sqrt4t^2+4t^4+1)

(=sqrt(2t^2+1)^2)

(=2t^2+1)

Tangent vectors:

(T(t)=fracR"(t)R"(t))

(=frac12t^2+1)

(T(-2)=frac12(-2)^2+1)

(T(-2)=frac12(-2)^2+1)

(=)

(T"(t)=)

(=)

(=)

(|T"(t)|=sqrtfrac(2-4t^2)^2+(4t)^2+(-4t)^2(2t^2+1)^4)

(=frac1(2t^2+1)^2sqrt4-16t^2+16t^4+16t^2+16t^2)

(=frac1(2t^2+1)^2sqrt16t^4+16t^2+4)

(=frac2(2t^2+1)(2t^2+1)^2)

(=frac22t^2+1)

The typical vectors.

You are watching: Find the vectors t, n, and b at the given point.

(N(t)=fracT"(t)T"(t)=frac1(2t^2+1)^2cdotfrac2t^2+12)

(=frac1(2t^2+1))

(N(-2)=frac1(2(-2)^2+1))

(=)

The binormal vector is,

(B(-2)=T(-2) imes N(-2))

(=eginvmatrixi&j&k\-frac49&frac89&frac19\-frac79&-frac49&frac49endvmatrix)

(=(frac3281+frac481)i-(-frac1681+frac781)j+(frac1681+frac5681)k)

(=)


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