If + - k &~ 2 -3 + k are adjacent sides of a parallelogram, then length of its diagonals are

Step-by-step solution

Let the adjacent sides of the parallelogram be represented by vectors a = i + j - k and b = 2 i - 3 j + k . The diagonals of the parallelogram are given by d₁ = a + b and d₂ = a - b . Calculating d₁ : d₁ = ( i + j - k ) + (2 i - 3 j + k ) = 3 i - 2 j + 0 k . The length of the first diagonal is | d₁ | = 3^2 + (-2)^2 + 0^2 = 9 + 4 = 13 . Calculating d₂ : d₂ = ( i + j - k ) - (2 i - 3 j + k ) = - i + 4 j - 2 k . The length of the second diagonal is | d₂ | = (-1)^2 + 4^2 + (-2)^2 = 1 + 16 + 4 = 21 . The lengths of the