# Suppose vector u =<3,-4,-2> and vector v = <-2,2,1>are two vectors that form the sides of a parallelogram. what are the lengths of the two diagonals of the parallelogram?

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You need to compose the vectors `bar u` and `bar v` to find the resultant vector that represents the first diagonal of parallelogram, such that:

`bar u + bar v = bar d_1`

`3 bar i - 4 bar j - 2 bar k - 2 bar i + 2 bar j + bar k = bar d_1`

Adding like terms yields:

`bar i - 2 bar j - bar k = bar d_1`

You need to evaluate the length of diagonal `d_1` such that:

`|bar d_1| = sqrt(1^2 + (-2)^2 + (-1)^2)`

`|bar d_1| = sqrt(6)`

You need to compose the vectors `bar u` and `bar v` to find the resultant vector that represents the second diagonal of parallelogram, such that:

`bar u + (- bar v) = bar d_2 `

`3 bar i - 4 bar j - 2 bar k + 2 bar i - 2 bar j - bar k = bar d_2`

`5 bar i - 6 bar j - 3 bar k = bar d_2`

You need to evaluate the length of diagonal `d_2` such that:

`|bar d_2| = sqrt(5^2 + (-6)^2 + (-3)^2)`

`|bar d_2| = sqrt(70)`

**Hence, evaluating the lengths of diagonal of parallelogram, under the given conditions, yields `|bar d_1| = sqrt(6)` and **`|bar d_2| = sqrt(70).`