Determine Whether the Point P Is in the Convex Hull

question 46

Multiple Choice

Determine whether the point p is in the convex hull of S.
- $$\begin{array} { l } \mathrm { S } = \left\{ \mathbf { v } _ { 1 } , \mathbf { v } _ { 2 } , \mathbf { v } _ { 3 } , \mathbf { v } _ { 4 } \right\} \\\mathbf { v } _ { 1 } = \left[ \begin{array} { l } 2 \\0 \\5 \\1\end{array} \right] , \mathbf { v } _ { 2 } = \left[ \begin{array} { r } 1 \\1 \\- 1 \\- 3\end{array} \right] , \mathbf { v } _ { 3 } = \left[ \begin{array} { r } 3 \\2 \\0 \\- 4\end{array} \right] , \mathbf { p } = \left[ \begin{array} { r } 0 \\- 1 \\4 \\2\end{array} \right]\end{array}$$

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

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Definitions:

Future Interest

A person’s present right to property ownership and possession in the future.

Conditional Estate

A form of property ownership that is subject to certain conditions or the occurrence of a specific event.

Mislaid Property

Property that is intentionally placed somewhere but then forgotten by the owner, distinct from lost or abandoned property.

Lost Property

Items that the owner has unintentionally left or dropped, and where the owner has forgotten or is unaware of their location.