Let e1 0 1 0 and e2 45 0 35 Find vector e3 such that e1 e2

Let e_1 = [0 1 0] and e_2 = [4/5 0 3/5]. Find vector e_3 such that {e_1, e_2, e_3} form an orthonormal basis. How many different choices for e_3 there is? Express vector b = [1 2 3] as a linear combination of vectors e_1, e_2, e_3.

Solution

a)

Let, e3=[x y z]

e3.e1=0 gives y=0

e3.e2=0 gives

4x+3z=0

x=-3z/4

e3=[-3z/4 0 z]

|e3|=1=sqrt{9z^2/16+z^2}=zsqrt{25/16}

z=4/5 , -4/5

So, e3=[3/5 0 -4/5] , [-3/5 0 4/5]

b.

There are only 2 choices for e3 as we computed in part a.

c.

e3=[3/5 0 -4/5]

Let,

b=re1+se2+te3

So,s=2

1=4r/5+3t/5

4r+3t=5

3=3r/5-4t/5

3r-4t=15

Solving for r,t gives

r=13/5, t=-9/5

b=13e1/5+2e2-9e3/5