Q8 Find the volume of the parallelopied spanned by the vectors x~ = - 12i~ + 5j~

Question

Q8

Find the volume of the parallelopied spanned by the vectors x~ = - 12i~ + 5j~ - 4k~ , y = i - 8j + 7k and z = - 3i~ + 6j~ - 7k~. The volume of the parallelopied is (enter a number only) After multiplying through by the integrating factor, the linear differential equation, dy /dx + 2y = e4x, with initial condition y = - 5 when x = 0 , has the modified form d / dx((e2x) middot y) = (e2x) middot e4x. Give the exact answer to the appropriate definite integral of the Left Hand Side of the modified differential equation. The answer to the appropriate integral of the left hand side is .

Explanation / Answer

Volume = Magnitude of cross product of all 3 vectors = x X y X z

= |-1106i-271j+258k|= 1167.58 units

y = (e6x)/e2x = (1/6)(e6x+c)/e2x

given at y = -5 ,x = 0 ==> c = -31

==> va;ue of Left hand side of modefied equation = (1/6)(e6x-31) = -5

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