Let $C$ be a smooth curve defined by $\mathbf { r } ( t ) = 7 \mathbf { i } + t \mathbf { j } + 4 t ^ { 2 } \mathbf { k }$, and let $\mathrm { T } ( t )$ and $\mathrm { N } ( t )$ be the unit tangent vector and unit normal vector to $C$ corresponding to $t$. The plane determined by $\mathbf { T }$ and $\mathbf { N }$ is called the osculating plane. Find an equation of the osculating plane of the curve described by $\mathbf { r } ( t )$ at $t = 1$.

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The Answer of Let $C$ be a smooth curve defined by $\mathbf { r } ( t ) = 7 \mathbf { i } + t \mathbf { j } + 4 t ^ { 2 } \mathbf { k }$, and let $\mathrm { T } ( t )$ and $\mathrm { N } ( t )$ be the unit tangent vector and unit normal vector to $C$ corresponding to $t$. The plane determined by $\mathbf { T }$ and $\mathbf { N }$ is called the osculating plane. Find an equation of the osculating plane of the curve described by $\mathbf { r } ( t )$ at $t = 1$.

Let CCC Be a Smooth Curve Defined By r(t)=7i+tj+4t2k\mathbf { r } ( t ) = 7 \mathbf { i } + t \mathbf { j } + 4 t ^ { 2 } \mathbf { k }r(t)=7i+tj+4t2k

Let $C$ Be a Smooth Curve Defined By $\mathbf { r } ( t ) = 7 \mathbf { i } + t \mathbf { j } + 4 t ^ { 2 } \mathbf { k }$