Given the points \( A(-3,2), B(7,13), C(2,4), D(5,4), E(\sqrt{5}, \sqrt{5}) \), and \( F(4 \sqrt{5},-3 \sqrt{5}) \), find the position vector equal to the following vectors. \( \begin{array}{llll}\text { a. } \overrightarrow{A B} & \text { b. } \overrightarrow{A C} & \text { c. } \overrightarrow{E F} & \text { d. } \overrightarrow{C D}\end{array} \)

To find the position vectors for the given vectors, we will use the formula for the position vector from point \( P(x_1, y_1) \) to point \( Q(x_2, y_2) \): \[ \overrightarrow{PQ} = (x_2 - x_1, y_2 - y_1) \] Now, let's calculate each vector step by step. ### a. \( \overrightarrow{AB} \) Given points: - \( A(-3, 2) \) - \( B(7, 13) \) Using the formula: \[ \overrightarrow{AB} = (7 - (-3), 13 - 2) = (7 + 3, 11) = (10, 11) \] ### b. \( \overrightarrow{AC} \) Given points: - \( A(-3, 2) \) - \( C(2, 4) \) Using the formula: \[ \overrightarrow{AC} = (2 - (-3), 4 - 2) = (2 + 3, 2) = (5, 2) \] ### c. \( \overrightarrow{EF} \) Given points: - \( E(\sqrt{5}, \sqrt{5}) \) - \( F(4\sqrt{5}, -3\sqrt{5}) \) Using the formula: \[ \overrightarrow{EF} = (4\sqrt{5} - \sqrt{5}, -3\sqrt{5} - \sqrt{5}) = (4\sqrt{5} - 1\sqrt{5}, -4\sqrt{5}) = (3\sqrt{5}, -4\sqrt{5}) \] ### d. \( \overrightarrow{CD} \) Given points: - \( C(2, 4) \) - \( D(5, 4) \) Using the formula: \[ \overrightarrow{CD} = (5 - 2, 4 - 4) = (3, 0) \] ### Summary of Results The position vectors are: - \( \overrightarrow{AB} = (10, 11) \) - \( \overrightarrow{AC} = (5, 2) \) - \( \overrightarrow{EF} = (3\sqrt{5}, -4\sqrt{5}) \) - \( \overrightarrow{CD} = (3, 0) \)