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. $\mathbf { p } = 2 \mathbf { v } _ { 1 } + 3 \mathbf { v } _ { 2 } - 4 \mathbf { v } _ { 3 }$

Question

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. $\mathbf { p } = 2 \mathbf { v } _ { 1 } + 3 \mathbf { v } _ { 2 } - 4 \mathbf { v } _ { 3 }$

Quizplus · Accepted Answer

To determine if a point $p$ is in the convex hull of a set $S$, one must find a combination of the points in $S$ that can linearly represent $p$ with coefficients that are non-negative and sum up to 1. Given the options, none provide a valid combination of $\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3$ with non-negative coefficients that sum to 1 to represent $p$, indicating $p$ is not in the convex hull of $S$.

Determine Whether the Point P Is in the Convex Hull