Final answer:
The expansion of the vector v = <1,-1,3> in terms of the vectors u₁ = <-2,-3,-1>, u₂ = <-2,2,-2>, u₃ = <8,-2,-10> is -2u₁ - 10u₂ - 20u₃.
Step-by-step explanation:
To find the expansion of the vector v = <1,-1,3> in terms of the vectors u₁ = <-2,-3,-1>, u₂ = <-2,2,-2>, u₃ = <8,-2,-10>, we can use the dot product of v with each of the u vectors. Let's calculate each component of the expansion:
v · u₁ = (1)(-2) + (-1)(-3) + (3)(-1) = -2 + 3 - 3 = -2
v · u₂ = (1)(-2) + (-1)(2) + (3)(-2) = -2 - 2 - 6 = -10
v · u₃ = (1)(8) + (-1)(-2) + (3)(-10) = 8 + 2 - 30 = -20
Therefore, the expansion of v in terms of the u vectors is -2u₁ - 10u₂ - 20u₃.