Find the unit normal vector N(t) to the curve r (t) = $\left\langle t , 2 t , t ^ { 2 } \right\rangle$ when t = 1.
A) $\left\langle \frac { 1 } { 3 } , \frac { 2 } { 3 } , \frac { 2 } { 3 } \right\rangle$

B) $( 1,0,0 )$
C)$( 0,1,0)$
D) $\left( 0 , \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$
E) $\left( - \frac { 2 } { 3 \sqrt { 5 } } , - \frac { 4 } { 3 \sqrt { 5 } } , \frac { 5 } { 3 \sqrt { 5 } } \right)$
F) $\left( \frac { 1 } { \sqrt { 2 } } , 0 , \frac { 1 } { \sqrt { 2 } } \right)$

G) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$

H) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$

Question

Find the unit normal vector N(t) to the curve r (t) = $\left\langle t , 2 t , t ^ { 2 } \right\rangle$ when t = 1.
A) $\left\langle \frac { 1 } { 3 } , \frac { 2 } { 3 } , \frac { 2 } { 3 } \right\rangle$

B) $( 1,0,0 )$ 
C)$( 0,1,0)$ 
D) $\left( 0 , \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$ 
E) $\left( - \frac { 2 } { 3 \sqrt { 5 } } , - \frac { 4 } { 3 \sqrt { 5 } } , \frac { 5 } { 3 \sqrt { 5 } } \right)$ 
F) $\left( \frac { 1 } { \sqrt { 2 } } , 0 , \frac { 1 } { \sqrt { 2 } } \right)$

G) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , 0 \right)$

H) $\left( \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } , \frac { 1 } { \sqrt { 2 } } \right)$

Quizplus · Accepted Answer

To find the unit normal vector $N(t)$ to the curve $r(t) = \langle t, 2t, t^2 \rangle$ at $t = 1$, we first find the tangent vector $T(t)$ by differentiating $r(t)$ with respect to $t$, which gives $r'(t) = \langle 1, 2, 2t \rangle$. At $t = 1$, $r'(1) = \langle 1, 2, 2 \rangle$. The magnitude of $r'(1)$ is $\sqrt{1^2 + 2^2 + 2^2} = \sqrt{9} = 3$. The unit tangent vector $T(1)$ is then $\frac{1}{3}\langle 1, 2, 2 \rangle$. To find the normal vector, we need a vector that is orthogonal to $T(1)$. However, the question directly asks for the unit normal vector, which is a normalized version of any normal vector. The correct answer, after normalizing the derivative of the tangent vector or using other methods to find a perpendicular vector and normalizing it, is given by option E, which represents a correctly normalized vector orthogonal to the tangent vector at $t = 1$.

Find the Unit Normal Vector N(t) to the Curve R ⟨t,2t,t2⟩\left\langle t , 2 t , t ^ { 2 } \right\rangle⟨t,2t,t2⟩

Find the Unit Normal Vector N(t) to the Curve R $\left\langle t , 2 t , t ^ { 2 } \right\rangle$