The problem involves translating and reflecting a point geometrically.

1. The point B has coordinates (3,2).
2. It is translated using the vector $T = \begin{bmatrix} 3 \\ 1 \end{bmatrix}$.
3. Then it is reflected across the vertical line $x = 4$.

Step 1: Translation

To translate the point $B(3,2)$ by the vector $T = \begin{bmatrix} 3 \\ 1 \end{bmatrix}$:

• New $x$-coordinate: $3 + 3 = 6$
• New $y$-coordinate: $2 + 1 = 3$

After translation, the new point is $B'(6, 3)$.

Step 2: Reflection across $x = 4$

To reflect the point $B'(6, 3)$ across the line $x = 4$, we calculate the horizontal distance from the point to the line:

• Distance from $6$ to $4$ is $6 - 4 = 2$.
• To reflect, we subtract this distance from $4$, which gives $4 - 2 = 2$.

The $y$-coordinate remains the same during a reflection across a vertical line.

Thus, the final reflected point is $B''(2, 3)$.

Conclusion

None of the provided answers matches $B''(2, 3)$, which indicates a discrepancy with the given options. Would you like to further analyze this or explore related concepts?

Here are 5 related questions to expand your understanding:

1. What is the general formula for translating a point by a vector?
2. How do you reflect a point across a vertical line?
3. Can you explain how translation affects the geometry of shapes?
4. How would the reflection change if we reflected across the line $y = 4$ instead?
5. What are some real-world applications of point reflection and translation?

Tip: When reflecting a point across a vertical line, the $y$-coordinate stays the same while the $x$-coordinate changes symmetrically.