https://file.aiquickdraw.com/mathbot/file/1726108121404em696ope.jpg

The image provides a vector problem with a vector \(\vec{v}\) originating from the point \((2, 3)\) and ending at point \((x, y)\). The conditions given are:

\[
\vec{v} \cdot \vec{i} = \frac{1}{\sqrt{2}}, \quad \vec{v} \cdot \vec{j} = -\frac{1}{\sqrt{2}}
\]

where:
- \(\vec{i} = (1, 0)\) is the unit vector in the x-direction.
- \(\vec{j} = (0, 1)\) is the unit vector in the y-direction.

### Step-by-Step Solution

1. **Vector Components:**

   Let \(\vec{v} = (x - 2, y - 3)\), representing the vector from \((2, 3)\) to \((x, y)\).

2. **Dot Product Equations:**

   The dot products given are:
   
   \[
   \vec{v} \cdot \vec{i} = (x - 2) \cdot 1 + (y - 3) \cdot 0 = x - 2 = \frac{1}{\sqrt{2}}
   \]

   \[
   \vec{v} \cdot \vec{j} = (x - 2) \cdot 0 + (y - 3) \cdot 1 = y - 3 = -\frac{1}{\sqrt{2}}
   \]

3. **Solve for \(x\) and \(y\):**

   From the first equation:
   
   \[
   x - 2 = \frac{1}{\sqrt{2}} \implies x = 2 + \frac{1}{\sqrt{2}}
   \]

   From the second equation:
   
   \[
   y - 3 = -\frac{1}{\sqrt{2}} \implies y = 3 - \frac{1}{\sqrt{2}}
   \]

### Final Answer:

\[
x = 2 + \frac{1}{\sqrt{2}}, \quad y = 3 - \frac{1}{\sqrt{2}}
\]

Would you like more details or have any further questions? Here are some related topics you might find interesting:

1. What is a dot product, and how is it calculated geometrically?
2. How do vectors relate to coordinate geometry?
3. What are the properties of unit vectors?
4. How can the dot product be used to find angles between vectors?
5. What is the significance of the unit vectors \(\vec{i}\) and \(\vec{j}\) in a Cartesian plane?

**Tip:** Understanding the dot product's properties helps in various applications, including physics and engineering problems involving force and work.

Let \( \vec{v} \) be as depicted below. Suppose \( \vec{v} \cdot \vec{i} = \frac{1}{\sqrt{2}} \) and \( \vec{v} \cdot \vec{j} = -\frac{1}{\sqrt{2}} \). Calculate \( x \) and \( y \).

This problem involves calculating the coordinates of a vector \( \vec{v} \) originating at (2, 3) using the dot product. The given dot product conditions \( \vec{v} \cdot \vec{i} = \frac{1}{\sqrt{2}} \) and \( \vec{v} \cdot \vec{j} = -\frac{1}{\sqrt{2}} \) provide a step-by-step solution to find \( x \) and \( y \).

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