The problem asks to determine the value of $x$ so that the vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$ are collinear, where the points are:

• $A(-7, -8)$
• $B(-2, -4)$
• $C(-6, 4)$
• $D(4, x)$

Solution:

1. Find the coordinates of the vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$:

• $\overrightarrow{AB} = B - A = (-2 - (-7), -4 - (-8)) = (5, 4)$
• $\overrightarrow{CD} = D - C = (4 - (-6), x - 4) = (10, x - 4)$

2. Condition for collinearity:

Two vectors $(x_1, y_1)$ and $(x_2, y_2)$ are collinear if their components satisfy: $\frac{x_1}{x_2} = \frac{y_1}{y_2}$ This is equivalent to: $x_1 \cdot y_2 = x_2 \cdot y_1$

For $\overrightarrow{AB}$ and $\overrightarrow{CD}$, this becomes: $5 \cdot (x - 4) = 4 \cdot 10$

3. Solve the equation:

$5(x - 4) = 40$ $5x - 20 = 40$ $5x = 60$ $x = 12$

Final Answer:

The value of $x$ is 12.

Would you like me to provide more details or explore another related concept?
Here are 5 related questions to deepen your understanding:

1. How do you verify if two vectors are collinear geometrically?
2. What happens if $x \neq 12$ in this problem? Are the vectors parallel or not?
3. Can you generalize the collinearity condition to 3D vectors?
4. How does the direction ratio of the vectors change with different $x$?
5. What are the applications of vector collinearity in physics and engineering?

Tip: Always simplify vector expressions before testing conditions like collinearity—it reduces the chances of error!