The altitude of the parallelopiped, whose coterminous edges are the vectors a = i + j + k , b = 2 i + 4 j - k , c = i + j + 3 k , where a , b are the sides of the base of parallelopiped, is

Step-by-step solution

The volume of the parallelepiped is given by the scalar triple product V = |[ a b c ]| . V = vmatrix 1 & 1 & 1 2 & 4 & -1 1 & 1 & 3 vmatrix = 1(12 + 1) - 1(6 + 1) + 1(2 - 4) = 13 - 7 - 2 = 4 The area of the base formed by a and b is A = | a b | . a b = vmatrix i & j & k 1 & 1 & 1 2 & 4 & -1 vmatrix = i (-1 - 4) - j (-1 - 2) + k (4 - 2) = -5 i + 3 j + 2 k A = (-5)^2 + 3^2 + 2^2 = 25 + 9 + 4 = 38 The altitude h of the parallelepiped is the ratio of the volume to the area of the base. h = V A = 4 38 = 4 2 19 = 2 2 19