Let's solve this step by step as requested: --- ### **Step 1:** **Question:** "If \|u + v\| = \|u - v\|, then \|u + v\|² = \|u - v\|² and by the definition of length we obtain which of the following?" **Options:** - (u, v) = (v, u) - \|u\|² + 2(u, v) + \|v\|² = \|u\|² - 2(u, v) + \|v\|² - \|u\|² + (u, v) + (v, u) + \|v\|² = \|u\|² - (u, v) - (v, u) + \|v\|² - \|u\|² + 2(u, v) = \|u\|² - 2(u, v) **Explanation:** By definition, \|a\|² = (a, a). So, \[ \|u + v\|^2 = (u+v, u+v) = (u, u) + 2(u, v) + (v, v) = \|u\|^2 + 2(u, v) + \|v\|^2 \] \[ \|u - v\|^2 = (u-v, u-v) = (u, u) - 2(u, v) + (v, v) = \|u\|^2 - 2(u, v) + \|v\|^2 \] So the correct answer is: **\|u\|² + 2(u, v) + \|v\|² = \|u\|² - 2(u, v) + \|v\|²** --- ### **Step 2:** **Question:** "Using the properties of the inner product and the definition of length, the previous equation gives us which of the following?" **Options:** - (u, v) = (v, u) - 0 = \|u\|² + 2(u, v) + \|v\|² - (\|u\|² - 2(u, v) + \|v\|²) - 0 = \|u\|² - 2(u, v) + \|v\|² - (\|u\|² + 2(u, v) + \|v\|²) - 0 = (\|u\|² + \|v\|²) - (\|u\|² + \|v\|²) **Explanation:** Subtract the right side from the left side: \[ \|u\|^2 + 2(u, v) + \|v\|^2 - [\|u\|^2 - 2(u, v) + \|v\|^2] = 0 \] Which simplifies to: \[ 0 = \|u\|^2 + 2(u, v) + \|v\|^2 - (\|u\|^2 - 2(u, v) + \|v\|^2) \] So the correct answer is: **0 = \|u\|² + 2(u, v) + \|v\|² - (\|u\|² - 2(u, v) + \|v\|²)** --- ### **Final Answer:** Step 1: **\|u\|² + 2(u, v) + \|v\|² = \|u\|² - 2(u, v) + \|v\|²** Step 2: **0 = \|u\|² + 2(u, v) + \|v\|² - (\|u\|² - 2(u, v) + \|v\|²)**