Problem 3 · Use a zero-sum identity for cyclic differences · 3 points
Made by AI and not reviewed by a real person.
Real numbers a, b, and c satisfy a+b+c=4 and a²+b²+c²=14. If u=(a−b, b−c, c−a) and v=(b−c, c−a, a−b), what is u·v?
- −26
- 0
- 13
- −13
- −21
Show a hint
- Look for a relationship among the three coordinate differences before expanding the dot product.
- Let x=a−b, y=b−c, and z=c−a. Use (x+y+z)² to relate their pairwise products to their squares, then use the given information to find the sum of those squares.
Show the answer and solution
Answer: D — −13
Set x=a−b, y=b−c, and z=c−a. Since x+y+z=0, the dot product is xy+yz+zx=−(x²+y²+z²)/2. Also, x²+y²+z²=3(a²+b²+c²)−(a+b+c)²=42−16=26. Thus u·v=−13. The tempting −26 comes from finding the correct sum of squares but forgetting to divide by 2.