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:

Price Floor

A government or regulatory-imposed minimum price for a commodity or service, intended to prevent the market price from falling below a certain level.

Price Ceiling

A government-imposed limit on how high a price can be charged for a product, service, or commodity, usually intended to protect consumers.

Purely Competitive Market

A market structure characterized by a large number of small firms, a homogeneous product, free entry and exit, and perfect information.

Surplus

The situation in which the quantity of a good or service supplied exceeds the quantity demanded at the current price.