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. \( \begin{array}{llll}\text { a. } \overrightarrow{A B} & \text { b. } \overrightarrow{A C} & \text { c. } \overrightarrow{E F} & \text { d. } \overrightarrow{C D}\end{array} \)
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To find the position vectors for the vectors given, we can use the formula for finding the vector from point \(P(x_1, y_1)\) to point \(Q(x_2, y_2)\), which is given by: \[ \overrightarrow{PQ} = (x_2 - x_1, y_2 - y_1) \] Now, let's calculate each of the vectors: **a. \(\overrightarrow{A B}\)** Given points \(A(-3,2)\) and \(B(7,13)\), \[ \overrightarrow{AB} = (7 - (-3), 13 - 2) = (7 + 3, 11) = (10, 11) \] **b. \(\overrightarrow{A C}\)** Given points \(A(-3,2)\) and \(C(2,4)\), \[ \overrightarrow{AC} = (2 - (-3), 4 - 2) = (2 + 3, 2) = (5, 2) \] **c. \(\overrightarrow{E F}\)** Given points \(E(\sqrt{5}, \sqrt{5})\) and \(F(4\sqrt{5}, -3\sqrt{5})\), \[ \overrightarrow{EF} = (4\sqrt{5} - \sqrt{5}, -3\sqrt{5} - \sqrt{5}) = (4\sqrt{5} - 1\sqrt{5}, -4\sqrt{5}) = (3\sqrt{5}, -4\sqrt{5}) \] **d. \(\overrightarrow{C D}\)** Given points \(C(2,4)\) and \(D(5,4)\), \[ \overrightarrow{CD} = (5 - 2, 4 - 4) = (3, 0) \] ### Summary of Position Vectors: - \( \overrightarrow{AB} = (10, 11) \) - \( \overrightarrow{AC} = (5, 2) \) - \( \overrightarrow{EF} = (3\sqrt{5}, -4\sqrt{5}) \) - \( \overrightarrow{CD} = (3, 0) \)
