(Solved): 4. Suppose that \( A B \) is the diameter of a circle with center \( O \) and that \( C \) is a po ...

4. Suppose that \( A B \) is the diameter of a circle with center \( O \) and that \( C \) is a point on the circle, with \( C \) not equal to either \( A \) or \( B \). (See figure below.) Use the following steps to prove that the vectors \( \overrightarrow{C A} \) and \( \overrightarrow{C B} \) are orthogonal. (a) Why can you assume that the point \( O \) is the origin and that the line segment \( \overline{A B} \) is on the \( x \)-axis? (b) Suppose that angle \( \angle B O C=\theta \) and that the radius of the circle is \( R \). Find a formula for the vector \( \mathbf{c}=\overrightarrow{O C} \) in terms of \( R \) and \( \theta \). (c) Let \( \mathbf{a}=\overrightarrow{A C} \) and \( \mathbf{b}=\overrightarrow{B C} \). Use the triangle law for the addition of vectors to find formulae for the vectors \( \mathbf{a} \) and \( \mathbf{b} \) in terms of \( R \) and \( \theta \). 1 (d) Calculate the dot product of \( \mathbf{a} \) and \( \mathbf{b} \). What can you conclude? (e) What happens when \( C=A \) or \( C=B \) ?