Consider a parallelogram whose adjacent sides are represented by the vectors p = m i + 2 j - k and q = i - m j + 2 k , where m is a real number. If the area of the orthogonal projection of this parallelogram on the plane 2x - y + 2z = 5 is 1 square unit, then the sum of all possible positive values of m is

Step-by-step solution

The vector area of the parallelogram is given by A = p q . A = vmatrix i & j & k m & 2 & -1 1 & -m & 2 vmatrix A = i (4 - m) - j (2m + 1) + k (-m^2 - 2) The normal vector to the given plane 2x - y + 2z = 5 is n = 2 i - j + 2 k . The magnitude of the normal vector is | n | = 2^2 + (-1)^2 + 2^2 = 9 = 3 . The unit normal vector is n = 2 i - j + 2 k 3 . The area of the orthogonal projection of the parallelogram on the plane is given by | A n | . A n = 2(4 - m) - 1(-2m - 1) + 2(-m^2 - 2) 3 A n = 8 - 2m + 2m + 1 - 2m^2 -