The document provides a step-by-step solution to finding a basis for the plane x1+4x2-5x3=0 in R3. It first defines the normal vector n = <1,4,-5> and explains that a basis consists of two vectors such that the dot product of each with n is 0. It then finds that the vectors n1 = <0,5,4> and n2 = <5,0,1> satisfy this requirement, and therefore form a basis for the given plane.

1. Hi,
Please, need help, and step by step. More detail. I need to understand.
Thank You
Find a basis for the plane
x1+4x2-5x3=0 in R^(3)
Solution
x 1 +4x 2 -5x 3 = 0.......(1)
n = <1,4,-5>
We have to find a basis for the plane (1). That is we have to find two vectors such that dot
product of each vector with n is 0.
Let n 1 = <0,5,4>
n 2 = <5,0,1>
n.n 1 = (1*0) + (4*5) + (-5*4) = 20-20 = 0
n.n 2 = (1*5) + (4*0) + (-5*1) = 5-5 = 0
Thus the vectors n 1 = <0,5,4> and n 2 = <5,0,1> forms a basis for the plane (1).