The problem asks us to translate each point on the coordinate grid by the given vector. Part a) involves translating the point (1, 1) by the vector $\begin{pmatrix} -3 \\ -2 \end{pmatrix}$. Part b) involves translating the point (2, -1) by the vector $\begin{pmatrix} 2 \\ -1 \end{pmatrix}$.

1. Problem Description

The problem asks us to translate each point on the coordinate grid by the given vector. Part a) involves translating the point (1, 1) by the vector $\begin{pmatrix} -3 \\ -2 \end{pmatrix}$. Part b) involves translating the point (2, -1) by the vector $\begin{pmatrix} 2 \\ -1 \end{pmatrix}$.

2. Solution Steps

a) The original point is (1, 1). The translation vector is $\begin{pmatrix} -3 \\ -2 \end{pmatrix}$. The translated point is obtained by adding the vector to the original point's coordinates.

$x_{new} = x_{old} + x_{vector} = 1 + (-3) = -2$

$y_{new} = y_{old} + y_{vector} = 1 + (-2) = -1$

So the new point is (-2, -1).

b) The original point is (2, -1). The translation vector is $\begin{pmatrix} 2 \\ -1 \end{pmatrix}$.

$x_{new} = x_{old} + x_{vector} = 2 + 2 = 4$

$y_{new} = y_{old} + y_{vector} = -1 + (-1) = -2$

So the new point is (4, -2).

3. Final Answer

a) The translated point is (-2, -1).

b) The translated point is (4, -2).