Find the area of the parallelogram whose diagonals are 3î + ) - 2k. and î - 3) + 4k.

The area of a parallelogram given two diagonals (d1 and d2) can be calculated using the formula:

Area = 1/2 * |d1 x d2|

where |d1 x d2| is the magnitude of the cross product of the two diagonals.

Given the diagonals d1 = 3i + j - 2k and d2 = i - 3j + 4k, we first need to find their cross product.

d1 x d2 = (3i + j - 2k) x (i - 3j + 4k)

= [(3)(-3) - (-2)(4)]i - [(3)(4) - (-2)(1)]j + [(3)(-3) - (1)(1)]k

= -5i - 14j - 10k

Then, we find the magnitude of this cross product:

|d1 x d2| = sqrt((-5)^2 + (-14)^2 + (-10)^2)

= sqrt(25 + 196 + 100)

= sqrt(321)

Finally, we substitute this into the formula for the area of the parallelogram:

Area = 1/2 * sqrt(321)

So, the area of the parallelogram is 1/2 * sqrt(321) square units.