Given r=(1+t2)i^+(4t−3)j^+(2t2−6t)k^r =(1+t^2)\hat{i} + (4t-3)\hat{j} + (2t^2-6t)\hat{k}r=(1+t2)i^+(4t−3)j^+(2t2−6t)k^
differentiating it with respect to t,
r′=2ti^+4j^+(4t−6)k^r' = 2t\hat{i} + 4\hat{j} + (4t-6)\hat{k}r′=2ti^+4j^+(4t−6)k^
unit tangent vector,
r′^=r′⃗∣r′∣=2ti^+4j^+(4t−6)k^4t2+16+(4t−6)2\hat{r'} = \frac{\vec{r'}}{|r'|} = \frac{2t\hat{i} + 4\hat{j} + (4t-6)\hat{k}}{\sqrt{4t^2 + 16 + (4t-6)^2}}r′^=∣r′∣r′=4t2+16+(4t−6)22ti^+4j^+(4t−6)k^ =2ti^+4j^+(4t−6)k^(20t2−12t+52)=ti^+2j^+(2t−3)k^(5t2−3t+13)=\frac{2t\hat{i} + 4\hat{j} + (4t-6)\hat{k}}{\sqrt{(20t^2-12t+52)}} = \frac{t\hat{i} + 2\hat{j} + (2t-3)\hat{k}}{\sqrt{(5t^2-3t+13)}}=(20t2−12t+52)2ti^+4j^+(4t−6)k^=(5t2−3t+13)ti^+2j^+(2t−3)k^
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