a Find two vectors u and w that are not scalar multiples of

a) Find two vectors u and w that are not scalar multiples of one another and are both orthogonal to the vector v = [2;2 ;3 ]

b) Using your answer to part a) find a system of 2 equations in 3 variables whose solution set is the line with parametric form [1 ;3; 2 ] + t [2 ;2; 3 ]

Solution

Let w =< x, y, z >, and we want

< 2; 2; 3 > · < x, y, z > = 2 x +2y + 3z = 0

That gives us two free parameters to work with - let y and z be arbitrary, and then x = -y (3/2)z.

Pick values for y and z; say y = 1, z = 1, and so x = (5/2). The vector < 5/2, 1, 1 > is orthogonal to < 2; 2; 3 >.

Or, let y = 1, z = 2, and so x = 4. The vector < 4, 1, 2 > is orthogonal to < 2, 2, 3 >.

Note that < 5/2, 1, 1 > and < 4, 1, 2 > are not scalar multiples of each other.

Your answers will probably vary, as there are an infinite number of choices.