Let a parallelogram have adjacent sides represented by the vectors u = 2 i + j - k and v = 3 i - j + k , where > 0 . If the diagonals of this parallelogram are mutually perpendicular, and the vector u is perpendicular to i + j + 5 k , then the value of u v is equal to

Step-by-step solution

Since u is perpendicular to i + j + 5 k , their dot product is zero: u ( i + j + 5 k ) = 0 (2)(1) + ( )(1) + (-1)(5) = 0 = 3 Thus, u = 2 i + 3 j - k . The squared magnitude of u is: | u |^2 = 2^2 + 3^2 + (-1)^2 = 4 + 9 + 1 = 14 The diagonals of the parallelogram are mutually perpendicular, which implies that the parallelogram is a rhombus. Hence, the adjacent sides must have equal magnitudes: | u | = | v | | u |^2 = | v |^2 The squared magnitude of v is: | v |^2 = 3^2 + (-1)^2 + ^2 = 10 + ^2 Equating the squared ma