Question 1

From |a+b|²=21 and |a-b|²=9, I get |a|²+|b|²=15 and a·b=3. Using the perpendicularity condition a·(a-b)=0, I find |a|²=3 and |b|²=12. The triangle OPR has area (1/2)|a×b|, and computing |a×b|²=|a|²|b|²-(a·b)²=3. So |a×b|²=36-9=27, giving |a×b|=3√3, and the area is 3√3/2 — matching option C. |a+b|² = 21 and |a−b|² = 9 give |a|² + |b|² = 15 and a·b = 3. a ⊥ (a−b) ⇒ |a|² = a·b = 3, so |b|² = 12. Area of triangle OPR = ½|a × (a+b)| = ½|a × b|, and |a × b|² = |a|²|b|² − (a·b)² = 36 − 9 = 27. Area = ½·3√3 = 3√3/2.

**Solving the equation** I need to solve some equations involving vectors and lengths. It looks like I'm working with two vectors, a and b, where c is perpendicular to a. I’ve got a relationship between them: a + b = 2a - c. I need to express everything in a way that relates back to norms and areas. The user seems to want the answer in JSON format, so I'll make sure to end with that.

**Calculating triangle area** I'm working on computing the area of a triangle with vertices O, P=a, and R=a+b. The area formula involves the cross product of vectors. Given the equations, I can find the necessary magnitudes and their relations. After breaking down the calculations, I'll arrive at an area result of 3√3/2. It seems like I need to verify the final details and produce results in JSON format.

Using perpendicularity, a·(a-b)=0 gives |a|²=a·b. Setting up the two magnitude equations and subtracting gives a·b=3, so |a|²=3. Adding the equations instead gives |a|²+|b|² = 15, so |b|²=12. For the triangle area with O at origin, P=a, R=a+b, the area equals half the magnitude of a×b. Computing |a×b|² via |a|²|b|²-(a·b)² gives 27, so |a×b|=3√3, making the area 3√3/2 — matching answer choice C.