There are a set of three problems in the book by: Henry Ricardo, titled: A Modern Introduction to Linear Algebra, in Chap. $1$, sec. $1.1$, part $B$. The last two problems (Q.$5$, Q.$6$) are dependent on the first problem (Q.$4$).
I expect that solution to the first problem would pave the way for the last two too.
Q.$4$.
Let $x = \begin{bmatrix}x_1\\x_2 \end{bmatrix}$. Define $x \ge 0$ to mean $x_1 \ge 0$ and $x_2 \ge 0$.
Define $x \le 0$ analogously. If $(x_1 + x_2)x \ge 0$, what must be true of $x$?
Soln. attempt:
$(x_1 + x_2)x \ge 0\implies(x_1 + x_2)\begin{bmatrix}x_1\\x_2 \end{bmatrix}\ge 0\implies \begin{bmatrix}x_1(x_1+x_2)\\x_2(x_1+x_2) \end{bmatrix}\ge 0\implies x_1(x_1+x_2)\ge 0, x_2(x_1+x_2)\ge 0$.
Leads to $6$ combination of possibilities for sign of values of $x_1, x_2, x_1+x_2$
If signs of $x_1, x_2$ are the same:
(i) $x_1\ge 0, x_2\ge 0$:
(ii) $x_1\le 0, x_2\le 0$:
If signs of $x_1, x_2$ are different:
Case specific combinations:
a. $x_1(x_1+x_2)$:
(iii)$x_1\ge 0, x_2\le 0, x_1+x_2\ge 0$:
(iv) $x_1\le 0, x_2\ge 0, x_1+x_2\le 0$:
b. $x_2(x_1+x_2)$:
(v)$x_2\ge 0, x_1\le 0, x_1+x_2\ge 0$:
(vi) $x_2\le 0, x_1\ge 0, x_1+x_2\le 0$:
Am unable to draw any inference that might be useful for solving the next two question below.
Q.$5$. Using the definition in the previous exercise, define $u \ge v$ to
mean $u- v \ge 0$, where $u$ and $v$ are vectors having the same
number of components. Consider the following vectors:
$x= \begin{bmatrix}3\\5\\-1 \end{bmatrix}, y= \begin{bmatrix}6\\5\\6 \end{bmatrix}, u = \begin{bmatrix}0\\0\\-2 \end{bmatrix}, v = \begin{bmatrix}4\\2\\0 \end{bmatrix}$
a. Show that $x \ge u$.
b. Show that $v \ge u$.
c. Is there any relationship between $x$ and $v$?
d. Show that $y \ge x$, $y \ge u$, and $y \ge v$.
Q.$6$. Extending the definitions in Exercises B4 and B5 in the obvious way, prove the following results for vectors $x, y, z$, and $w$ in $R_n$:
a. If $x \le y$ and $y \le z$, then $x \le z$.
b. If $x \le y$ and $z \le w$, then $x + z\le y+w$.
c. If $x \le y$ and $\lambda$ is a non-negative real number, then $\lambda x \le \lambda y$.
d. If $x \le y$ and $\mu$ is a negative real number, then $\mu x \ge \mu y$.
Their relation is that Question $4$ first define the definition of $\ge$ for a vector so that we can write down $x \ge 0$ and we know it means entrywise nonnegative. Its use in other part of the question seems to only be using its definition of greater than equal to $0$.
In Question $5$, we then enable comparison for some of the vectors, so that $u \ge v$ means elementwise comparison and verifies that it is not a total order.
Question $6$ studies some of the property of $\ge$.
More on question $4$:
More generally, if we have $$\left( \sum_{i=1}^n x_i\right)x \ge 0$$
$\sum_{i=1}^n x_i = 0$ satisfies the condition.
If $\sum_{i=1}^n x_i \ne 0$, we can divide it on both sides and use $6c$ and $6d$ and some verification to conclude the following result.
$$\left[ \sum_{i=1}^n x_i =0 \right] \lor [x \ge 0] \lor [x \le 0]$$