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It is possible if the size of the vector is 3 or larger. For example

\begin{align} a &= (-1, 1, 1)\\ b &= (1, -9, -3)\\ c &= (2, 3, -1)\\ \end{align}

The correlations are \begin{equation} \text{cor}(a,b) = -0.80...\\ \text{cor}(a,c) = -0.27...\\ \text{cor}(b,c) = -0.34... \end{equation}

We can prove that for vectors of size 2 this is not possible: \begin{align} \text{cor}(a,b) &< 0\\[5pt] 2\Big(\sum_i a_i b_i\Big) - \Big(\sum_i a_i\Big)\Big(\sum_i b_i\Big) &< 0\\[5pt] 2(a_1 b_1 + a_2 b_2) - (a_1 + a_2)(b_1 b_2) &< 0\\[5pt] 2(a_1 b_1 + a_2 b_2) - (a_1 + a_2)(b_1 b_2) &< 0\\[5pt] 2(a_1 b_1 + a_2 b_2) - a_1 b_1 + a_1 b_2 + a_2 b_1 + a_2 b_2 &< 0\\[5pt] a_1 b_1 + a_2 b_2 - a_1 b_2 + a_2 b_1 &< 0\\[5pt] a_1 (b_1-b_2) + a_2 (b_2-b_1) &< 0\\[5pt] (a_1-a_2)(b_1-b_2) &< 0 \end{align}

The formula makes sense: if $a_1$ is larger than $a_2$, $b_1$ has to be larger than $b_1$ to make the correlation negative.

Similarly for correlations between (a,c) and (b,c) we get

\begin{equation} (a_1-a_2)(c_1-c_2) < 0\\ (b_1-b_2)(c_1-c_2) < 0\\ \end{equation}

Clearly, all of these three formulas can not hold in the same time.

It is possible if the size of the vector is 3 or larger. For example

\begin{align} a &= (-1, 1, 1)\\ b &= (1, -9, -3)\\ c &= (2, 3, -1)\\ \end{align}

The correlations are \begin{equation} \text{cor}(a,b) = -0.80...\\ \text{cor}(a,c) = -0.27...\\ \text{cor}(b,c) = -0.34... \end{equation}

We can prove that for vectors of size 2 this is not possible: \begin{align} \text{cor}(a,b) &< 0\\[5pt] 2\Big(\sum_i a_i b_i\Big) - \Big(\sum_i a_i\Big)\Big(\sum_i b_i\Big) &< 0\\[5pt] 2(a_1 b_1 + a_2 b_2) - (a_1 + a_2)(b_1 b_2) &< 0\\[5pt] 2(a_1 b_1 + a_2 b_2) - (a_1 + a_2)(b_1 b_2) &< 0\\[5pt] 2(a_1 b_1 + a_2 b_2) - a_1 b_1 + a_1 b_2 + a_2 b_1 + a_2 b_2 &< 0\\[5pt] a_1 b_1 + a_2 b_2 - a_1 b_2 + a_2 b_1 &< 0\\[5pt] a_1 (b_1-b_2) + a_2 (b_2-b_1) &< 0\\[5pt] (a_1-a_2)(b_1-b_2) &< 0 \end{align}

The formula makes sense: if $a_1$ is larger than $a_2$, $b_2$ has to be larger than $b_1$ to make the correlation negative.

Similarly for correlations between (a,c) and (b,c) we get

\begin{equation} (a_1-a_2)(c_1-c_2) < 0\\ (b_1-b_2)(c_1-c_2) < 0\\ \end{equation}

Clearly, all of these three formulas can not hold at the same time.

It is possible if the size of the vector is 3 or larger. For example

\begin{equation} a = (-1, 1, 1)\\ b = (1, -9, -3)\\ c = (2, 3, -1)\\ \end{equation} The correlations are \begin{equation} \text{cor}(a,b) = -0.80...\\ \text{cor}(a,c) = -0.27...\\ \text{cor}(b,c) = -0.34... \end{equation}

We can prove that for vectors of size 2 this is not possible: \begin{equation} \text{cor}(a,b) < 0\\ 2(\sum_i a_i b_i) - (\sum_i a_i)(\sum_i b_i) < 0\\ 2(a_1 b_1 + a_2 b_2) - (a_1 + a_2)(b_1 b_2) < 0\\ 2(a_1 b_1 + a_2 b_2) - (a_1 + a_2)(b_1 b_2) < 0\\ 2(a_1 b_1 + a_2 b_2) - a_1 b_1 + a_1 b_2 + a_2 b_1 + a_2 b_2 < 0\\ a_1 b_1 + a_2 b_2 - a_1 b_2 + a_2 b_1 < 0\\ a_1 (b_1-b_2) + a_2 (b_2-b_1) < 0\\ (a_1-a_2)(b_1-b_2) < 0\\ \end{equation}

The formula makes sense: if $a_1$ is larger than $a_2$, $b_1$ has to be larger than $b_1$ to make the correlation negative.

Similarly for correlations between (a,c) and (b,c) we get

\begin{equation} (a_1-a_2)(c_1-c_2) < 0\\ (b_1-b_2)(c_1-c_2) < 0\\ \end{equation}

Clearly, all of these three formulas can not hold in the same time.

It is possible if the size of the vector is 3 or larger. For example

\begin{align} a &= (-1, 1, 1)\\ b &= (1, -9, -3)\\ c &= (2, 3, -1)\\ \end{align}

The correlations are \begin{equation} \text{cor}(a,b) = -0.80...\\ \text{cor}(a,c) = -0.27...\\ \text{cor}(b,c) = -0.34... \end{equation}

We can prove that for vectors of size 2 this is not possible: \begin{align} \text{cor}(a,b) &< 0\\[5pt] 2\Big(\sum_i a_i b_i\Big) - \Big(\sum_i a_i\Big)\Big(\sum_i b_i\Big) &< 0\\[5pt] 2(a_1 b_1 + a_2 b_2) - (a_1 + a_2)(b_1 b_2) &< 0\\[5pt] 2(a_1 b_1 + a_2 b_2) - (a_1 + a_2)(b_1 b_2) &< 0\\[5pt] 2(a_1 b_1 + a_2 b_2) - a_1 b_1 + a_1 b_2 + a_2 b_1 + a_2 b_2 &< 0\\[5pt] a_1 b_1 + a_2 b_2 - a_1 b_2 + a_2 b_1 &< 0\\[5pt] a_1 (b_1-b_2) + a_2 (b_2-b_1) &< 0\\[5pt] (a_1-a_2)(b_1-b_2) &< 0 \end{align}

The formula makes sense: if $a_1$ is larger than $a_2$, $b_1$ has to be larger than $b_1$ to make the correlation negative.

Similarly for correlations between (a,c) and (b,c) we get

\begin{equation} (a_1-a_2)(c_1-c_2) < 0\\ (b_1-b_2)(c_1-c_2) < 0\\ \end{equation}

Clearly, all of these three formulas can not hold in the same time.

It is possible if the size of the vector is 3 or larger. For example

\begin{equation} a = (-2, 7, 3)\\ b = (0, -4, -5)\\ c = (7, 1, 0)\\ \end{equation} The correlations are \begin{equation} \text{cor}(a,b) = -13\\ \text{cor}(a,c) = -7\\ \text{cor}(b,c) = -4 \end{equation}

We can prove that for vectors of size 2 this is not possible: \begin{equation} \text{cor}(a,b) < 0\\ 2(\sum_i a_i, b_i) - (\sum a)(\sum b) < 0\\ 2(a_1 b_1 + a_2 b_2) - (a_1 + a_2)(b_1 b_2) < 0\\ 2(a_1 b_1 + a_2 b_2) - (a_1 + a_2)(b_1 b_2) < 0\\ 2(a_1 b_1 + a_2 b_2) - a_1 b_1 + a_1 b_2 + a_2 b_1 + a_2 b_2 < 0\\ a_1 b_1 + a_2 b_2 - a_1 b_2 + a_2 b_1 < 0\\ a_1 (b_1-b_2) + a_2 (b_2-b_1) < 0\\ (a_1-a_2)(b_1-b_2) < 0\\ \end{equation}

The formula makes sense: if $a_1$ is larger than $a_2$, $b_1$ has to be larger than $b_1$ to make the correlation negative.

Similarly for correlations between (a,c) and (b,c) we get

\begin{equation} (a_1-a_2)(c_1-c_2) < 0\\ (b_1-b_2)(c_1-c_2) < 0\\ \end{equation}

Clearly, all of these three formulas can not hold in the same time.

It is possible if the size of the vector is 3 or larger. For example

\begin{equation} a = (-1, 1, 1)\\ b = (1, -9, -3)\\ c = (2, 3, -1)\\ \end{equation} The correlations are \begin{equation} \text{cor}(a,b) = -0.80...\\ \text{cor}(a,c) = -0.27...\\ \text{cor}(b,c) = -0.34... \end{equation}

We can prove that for vectors of size 2 this is not possible: \begin{equation} \text{cor}(a,b) < 0\\ 2(\sum_i a_i b_i) - (\sum_i a_i)(\sum_i b_i) < 0\\ 2(a_1 b_1 + a_2 b_2) - (a_1 + a_2)(b_1 b_2) < 0\\ 2(a_1 b_1 + a_2 b_2) - (a_1 + a_2)(b_1 b_2) < 0\\ 2(a_1 b_1 + a_2 b_2) - a_1 b_1 + a_1 b_2 + a_2 b_1 + a_2 b_2 < 0\\ a_1 b_1 + a_2 b_2 - a_1 b_2 + a_2 b_1 < 0\\ a_1 (b_1-b_2) + a_2 (b_2-b_1) < 0\\ (a_1-a_2)(b_1-b_2) < 0\\ \end{equation}

The formula makes sense: if $a_1$ is larger than $a_2$, $b_1$ has to be larger than $b_1$ to make the correlation negative.

Similarly for correlations between (a,c) and (b,c) we get

\begin{equation} (a_1-a_2)(c_1-c_2) < 0\\ (b_1-b_2)(c_1-c_2) < 0\\ \end{equation}

Clearly, all of these three formulas can not hold in the same time.