todai-math 2024 Q1

todai-math · Japan · science_official Vectors Introduction & 2D Angle or Cosine Between Vectors

Let $A(0, -1, 1)$ be a point in coordinate space. Suppose a point $P$ in the $xy$-plane satisfies all of the following conditions (i), (ii), (iii).

[(i)] $P$ is different from the origin $O$.
[(ii)] $\angle AOP \geq \dfrac{2}{3}\pi$
[(iii)] $\angle OAP \leq \dfrac{\pi}{6}$

Sketch the region that $P$ can occupy in the $xy$-plane.

Let $A(0, -1, 1)$ be a point in coordinate space. Suppose a point $P$ in the $xy$-plane satisfies all of the following conditions (i), (ii), (iii).

\begin{itemize}
\item[(i)] $P$ is different from the origin $O$.
\item[(ii)] $\angle AOP \geq \dfrac{2}{3}\pi$
\item[(iii)] $\angle OAP \leq \dfrac{\pi}{6}$
\end{itemize}

Sketch the region that $P$ can occupy in the $xy$-plane.

Paper Questions

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