To solve for the coordinates of point $C$ in the parallelogram $ABCD$ using vector equations, let's break down the problem step-by-step.

Given:

• The points $A(2, -1)$, $B(6, -3)$, and $D(3, 2)$ are corners of the parallelogram.
• The vector equation method relates the points through vector additions and subtractions.

Step 1: Understanding the Properties of a Parallelogram

In a parallelogram, opposite sides are parallel and of equal length. We know that:

1. The vector from $A$ to $B$ is the same as the vector from $D$ to $C$.
2. The vector from $A$ to $D$ is the same as the vector from $B$ to $C$.

Step 2: Write Vector Equation for $C$

We can use the vector property $\overrightarrow{AB} = \overrightarrow{DC}$ and $\overrightarrow{AD} = \overrightarrow{BC}$. Using this property, let's calculate the vectors and then solve for $C(x, y)$.

2.1 Find the Vector $\overrightarrow{AB}$:

$\overrightarrow{AB} = B - A = (6 - 2, -3 - (-1)) = (4, -2)$

2.2 Use the Parallelogram Property:

Since $\overrightarrow{AB} = \overrightarrow{DC}$, we can write the vector equation: $\overrightarrow{DC} = \overrightarrow{AB} = (4, -2)$ Therefore, the position vector of $C$ can be found by adding $\overrightarrow{DC}$ to point $D$: $C = D + \overrightarrow{DC} = (3, 2) + (4, -2) = (3 + 4, 2 - 2) = (7, 0)$

Thus, the coordinates of point $C$ are $C(7, 0)$.

Step 3: Verifying the Solution

To check if this is correct, we can verify the second property of the parallelogram: $\overrightarrow{AD} = \overrightarrow{BC}$.

• $\overrightarrow{AD} = (3 - 2, 2 - (-1)) = (1, 3)$
• $\overrightarrow{BC} = (7 - 6, 0 - (-3)) = (1, 3)$

Since both vectors are equal, the points $A, B, C, D$ indeed form a parallelogram, and $C(7, 0)$ is correct.

Final Answer:

The coordinates of point $C$ are $C(7, 0)$.

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Would you like to explore the solution further or have any questions? Here are some related topics you could look into:

1. How to find the area of a parallelogram using vectors.
2. How to use the midpoint formula in a parallelogram.
3. Understanding the vector cross product and its relation to parallelograms.
4. How to find the equation of a line through two points in vector form.
5. The concept of displacement vectors and their use in geometry.

Tip: In problems involving parallelograms, you can often use the properties of vector addition to solve for unknown points. Always look for equal or parallel vectors between opposite sides.