Determine Whether the Point P Is in the Convex Hull

question 23

Multiple Choice

Determine whether the point p is in the convex hull of S.
- $$\begin{array} { l } S = \left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } , \mathbf { v } _ { 4 } \right\} \\\mathbf { v } _ { 1 } = \left[ \begin{array} { l } 2 \\2 \\1\end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { r } 0 \\3 \\- 1\end{array} \right] , \mathbf { v } _ { 3 } = \left[ \begin{array} { r } 1 \\8 \\- 6\end{array} \right] , \mathbf { v } _ { 4 } = \left[ \begin{array} { r } - 1 \\2 \\4\end{array} \right] , \mathbf { p } = \left[ \begin{array} { l } 1 \\3 \\0\end{array} \right]\end{array}$$

• A) \(\mathbf { p } \in \operatorname { conv } \mathrm { S } . \mathbf { p } = \frac { 1 } { 2 } \mathbf { v } _ { 1 } + \frac { 1 } { 4 } \mathbf { v } _ { 2 } + \frac { 1 } { 4 } \mathbf { v } _ { 3 }\)
• B) \(\mathbf { p } \in \operatorname { conv }\) S. \(\mathbf { p } = \frac { 1 } { 2 } \mathbf { v } _ { 1 } + \frac { 1 } { 4 } \mathbf { v } _ { 2 } + \frac { 1 } { 8 } \mathbf { v } _ { 3 } + \frac { 1 } { 8 } \mathbf { v } _ { 4 }\)
• C) \(\mathbf { p } \in\) conv S. \(\mathbf { p } = 2 \mathbf { v } _ { 1 } + 3 \mathbf { v } _ { 2 } - 2 \mathbf { v } _ { 3 } - 2 \mathbf { v } _ { 4 }\)
• D) \(\mathbf { p } \notin\) conv \(\mathrm { S }\).

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Definitions:

Coleoptera

An order of insects that includes beetles and weevils, characterized by hard wing cases and a chewing mouthpart, representing the largest order of insects.

Insecta

The class within the phylum Arthropoda that includes insects, characterized by a three-part body, compound eyes, and one pair of antennae.

Biogeographic Province

A region defined by unique ecological and evolutionary processes that shape the distribution and diversity of species within it.

Marsupials

A group of mammals characterized by giving birth to relatively undeveloped offspring that typically continue to grow inside the mother's pouch.