Final answer:
The vectors ⟨−8,b,4⟩ and ⟨b,b²,b⟩ are orthogonal when their dot product equals zero, which occurs for b = 0 and b = 2.
Step-by-step explanation:
To determine for what values of b the vectors ⟨−8,b,4⟩ and ⟨b, b², b⟩ are orthogonal, we calculate the dot product of the two vectors. Two vectors are orthogonal if their dot product is equal to zero. The dot product is calculated as:
(-8) × (b) + (b) × (b²) + (4) × (b) = 0
-8b + b^3 + 4b = 0
b^3 - 4b = 0
b(b^2 - 4) = 0
b(b - 2)(b + 2) = 0
The values of b that satisfy this equation and therefore make the vectors orthogonal are b = 0, b = 2, and b = -2. However, since -2 is not an option in the given choices, the correct answers from the given choices are a) b=0 and b) b=2.