Question

sketch the curve with the given vector equation. Indicate with arrows the direction in which t increases:
r(t)=costi - costj + sintk

Answer 1

\[\vec{r}(t)=\cos{t}\cdot\vec{i}-\cos{t}\cdot\vec{j}+\sin{t}\cdot\vec{k}\]
\[\vec{r}(t)=\cos{t}\cdot(\vec{i}-\vec{j})+\sin{t}\cdot\vec{k}=
\cos{t}\cdot\vec{l}+\sin{t}\cdot\vec{k}\quad,\quad\vec{l}=\vec{i}-\vec{j}\]
\[\vec{l}=\sqrt{2}\cdot\vec{l}_0\quad,\quad |\vec{l}_0|=1\quad,\quad\vec{l}_0=\frac{\vec{i}-\vec{j}}{\sqrt{2}}\]
\[\vec{r}(t)=\sqrt{2}\cos{t}\cdot\vec{l}_0+\sin{t}\cdot\vec{k}\]
Curve is ellipse: center=(0,0,0) , (a,b)=(sqrt(2),1) in plane y=-x