Answer
$\dfrac{\sqrt{30}}{18}$
Work Step by Step
As we are given that $r(t)=\lt t^2, \ln t, t \ln t \gt$
Write Theorem 10. $\dfrac{|r'(t) \times r''(t)|}{|r'(t)|^3}$
Solve $r'(t)=\lt 2t, \frac{1}{t}, \ln t +1 \gt$; $r''(t)=\lt 2, -\frac{1}{t^2}, \frac{1}{t} \gt$
Thus,
$\dfrac{|r'(t) \times r''(t)|}{|r'(t)|^3}=\dfrac{|\frac{1}{t^2}(2+\ln t)i+2 \ln t j4/tk^ |}{[4t^2+\frac{1}{t^2}+ (\ln t+1)^2]^{3/2}}$
From the question, we have $(1,0,0)$
$\dfrac{|r'(t) \times r''(t)|}{|r'(t)|^3}=\dfrac{|2 i+0 j-4k |}{[6]^{3/2}}$
$=\dfrac{\sqrt{30}}{18}$
This is the desired result.
