Bextopa opromopaniponantinte (} -(2)/(sqrt (5)) (1)/(sqrt (5)) ) (c) (} -(1)/(sqrt (5)) (1)/(sqrt (5)) ) (} -(5)/(sqrt (5)) (1)/(sqrt (5)) ) (} -(2)/(sqrt (5)) (1)/(sqrt (5)) )

To solve the given problem, we need to perform the dot product of two vectors. The dot product of two vectors is calculated by multiplying the corresponding components of the vectors and then summing the results.<br /><br />Given vectors:<br />\[<br />\mathbf{A} = \left( -\frac{2}{\sqrt{5}}, \frac{1}{\sqrt{5}} \right)<br />\]<br />\[<br />\mathbf{B} = \left( \frac{2}{\sqrt{5}}, \frac{1}{\sqrt{5}} \right)<br />\]<br /><br />The dot product \(\mathbf{A} \cdot \mathbf{B}\) is calculated as follows:<br />\[<br />\mathbf{A} \cdot \mathbf{B} = \left( -\frac{2}{\sqrt{5}} \right) \cdot \left( \frac{2}{\sqrt{5}} \right) + \left( \frac{1}{\sqrt{5}} \right) \cdot \left( \frac{1}{\sqrt{5}} \right)<br />\]<br /><br />Now, let's calculate each term separately:<br />\[<br />\left( -\frac{2}{\sqrt{5}} \right) \cdot \left( \frac{2}{\sqrt{5}} \right) = -\frac{4}{5}<br />\]<br />\[<br />\left( \frac{1}{\sqrt{5}} \right) \cdot \left( \frac{1}{\sqrt{5}} \right) = \frac{1}{5}<br />\]<br /><br />Adding these results together gives:<br />\[<br />\mathbf{A} \cdot \mathbf{B} = -\frac{4}{5} + \frac{1}{5} = -\frac{3}{5}<br />\]<br /><br />So, the dot product of the given vectors is:<br />\[<br />-\frac{3}{5}<br />\]