https://file.aiquickdraw.com/mathbot/file/1725645522061vmezdu7h.jpg

In the image, we see a geometric problem involving the translation of a point in 3D space using a vector.

### Breakdown:

1. **Vector Translation:**
   The vector \(\vec{v} = \binom{2}{1}{3}\) (or written as \( \vec{v} = \begin{pmatrix} 2 \\ 1 \\ 3 \end{pmatrix} \)) is given. This vector describes a movement from point \( A(1|1|3) \).

   The vector specifies:
   - A translation of **2 units** in the \(x\)-direction,
   - **1 unit** in the \(y\)-direction,
   - **3 units** in the \(z\)-direction.

2. **Point Translation:**
   The point \( A(1|1|3) \) is translated by the vector \( \vec{v} \).

   - Original point \( A(1|1|3) \).
   - After translation by \( \vec{v} \), the new point (or "image point") is:
   \[
   A' = (1 + 2 | 1 + 1 | 3 + 3) = (3 | 2 | 6)
   \]
   So, the new position of the point is \( A'(3|2|6) \).

3. **Explanation of Vectors:**
   The note further explains that the vectors between the corresponding original points and their image points (after translation) all have the same magnitude and direction as the vector \( \vec{v} \). This makes sense geometrically, as translation by a vector does not change the relative distances between points, only their positions.

4. **Vector Illustration:**
   The small diagram shows the vector's effect, illustrating how a point is shifted in the three-dimensional space. The yellow area might indicate the area of translation.

Would you like more details on vector translation or how to apply this concept? 

### Follow-up Questions:
1. How do you calculate the magnitude of a vector in 3D space?
2. What are the general properties of translation in geometry?
3. How can you apply vector addition in different contexts like physics or navigation?
4. Can you explain how direction cosines are related to vector directions?
5. How do you translate multiple points by the same vector in 3D?

**Tip:** Always remember that vector translation preserves the shape and orientation of objects but moves them by the same vector distance.

Understand how to translate a point in 3D space using vector \( \vec{v} = \binom{2}{1}{3} \). The original point A(1|1|3) is translated by the vector, resulting in a new point A'(3|2|6). This concept highlights key vector translation principles, such as maintaining the shape and orientation during shifts.

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