Calculus Subspace

Question

Let the line L be spanned by -5,1,3 Find a basis of the orthogonal complement L⊥ of L

Answer 1

The line L is spanned by the vector (-5, 1, 3). The orthogonal complement is composed of all vectors (x, y, z) to satisfy the dot product equation: (-5, 1, 3)*(x, y, z) = -5x + y + 3z = 0. Now, replace x with 0, y with -3, and z with 1. This gives you -5(0) -3 + 3(1) = 3 - 3 = 0. You can also replace x with 1, y with 5, and z with 0, leaving the equation as 5 - 5 = 0. Thus, the basis for the orthogonal complement is 〈(1, 5, 0), (0, -3, 1)〉.