1.
Let u = [ 1, 1, 4 ] and v = [ 3, 2, 4 ].
(a) Find a vector n = [ x, y, z ] that is orthogonal to u and orthogonal to v. Hint: Use the dot
product to create a system of linear equations, then use Gauss-Jordan to solve that SLE.
(b) Use your answer in (a) to find the normal and general forms of the plane x = [3, 1, 1]+su+tv.
Solution
a) the dot product is multiplying and adding the components of each vector. for a vector to be
orthogonal to another vector the dot product has to equal zero. (1x+1y+4z)=0 any point on it is
orthogonal (3x+2y+4z)=0 1x=3x 1y=2y 4z=4z so a vector orthogonal would be [1,1,2], [1,1,3],
any value for the z will work.