Take V R2 with the inner product defined as 2x1x23y1y2 Find

Take V = R^2 with the inner product defined as =2x_1x_2+3y_1y_2 Find the magnitude of the vectors Find an orthogonal vector to each of the vectors u, ? and w from a).

Solution

(a)

u.u=2*1^2=2

Hence, ||u||=\\sqrt{2}

v.v=3*1^2=3

Hence, ||v||=\\sqrt{3}

w.w=2*3^2+3*2^2=18+12=30

||w||=sqrt{30}

(b)

Let, u.v=2*1*0+3*0*1=0

So, vectors orthogonal to u is v

vector orthogonal to v is u

Let,

z=(x,y)^t

w.z=0

2*3x+3*2y=0

x+y=0

So, z=(1,-1) is one vector orthogonal to w