We are tasked to find the coordinates of point $F(x, y)$ given the following:

1. $D = (1, 1)$ and $E = (2, 3)$ in Cartesian coordinates from $i+j$ and $2i+3j$, respectively.
2. Point $F$ lies on the line $y = -2x - 3$.
3. $\overrightarrow{DE}$ and $\overrightarrow{DF}$ are perpendicular.

Step-by-Step Solution:

Step 1: Calculate the slope of $\overrightarrow{DE}$

The slope $m_{DE}$ of line $DE$ is given by: $m_{DE} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{3 - 1}{2 - 1} = 2.$

Step 2: Perpendicularity condition

The slope of $\overrightarrow{DF}$, say $m_{DF}$, must satisfy the perpendicularity condition: $m_{DE} \cdot m_{DF} = -1.$ Substituting $m_{DE} = 2$: $2 \cdot m_{DF} = -1 \implies m_{DF} = -\frac{1}{2}.$

Thus, the slope of $\overrightarrow{DF}$ is $-\frac{1}{2}$.

Step 3: Parametric equation for $DF$

The slope $m_{DF} = -\frac{1}{2}$ implies: $y - 1 = -\frac{1}{2}(x - 1) \implies y = -\frac{1}{2}x + \frac{3}{2}.$ Point $F(x, y)$ lies on this line.

Step 4: Intersection of $DF$ and $y = -2x - 3$

Point $F$ also lies on the line $y = -2x - 3$. Solve the system of equations:

1. $y = -\frac{1}{2}x + \frac{3}{2}$,
2. $y = -2x - 3$.

Equating the two: $-\frac{1}{2}x + \frac{3}{2} = -2x - 3.$ Multiply through by 2 to eliminate fractions: $-x + 3 = -4x - 6.$ Simplify: $3x = -9 \implies x = -3.$

Substitute $x = -3$ into $y = -2x - 3$: $y = -2(-3) - 3 = 6 - 3 = 3.$

Step 5: Verify the solution

The coordinates of $F$ are $(-3, 3)$. To verify:

• $F(-3, 3)$ lies on $y = -2x - 3$: $3 = -2(-3) - 3$.
• The slope of $DF$ is: $m_{DF} = \frac{3 - 1}{-3 - 1} = \frac{2}{-4} = -\frac{1}{2}.$ This satisfies the perpendicularity condition.

Final Answer:

The coordinates of point $F$ are: $\boxed{(-3, 3)}.$

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Let me know if you need further clarification or details! Here are some related questions:

1. How to find equations of perpendicular lines in general?
2. How to determine if a point lies on a given line?
3. What is the geometric interpretation of perpendicular vectors?
4. How to calculate slopes using vector components?
5. How to solve systems of linear equations efficiently?

Tip: Always verify your solution by substituting back into the given conditions to ensure correctness!