Express the vector u below as a sum of two vectors u1 and u2

Express the vector u below as a sum of two vectors u₁ and u₂, where u₁ is parallel to the vector v given below, and u₂ is perpendicular to v. Make sure that the first vector in your sum is u₁ and the second is u₂. Use the square root symbol \'\' where needed to give an exact value for your answer.

v= [3,-1,-3]

u= [3,1,-2]

Solution

u =u1+u2

u1= ai+bj+ck and u2 =pi+qj+rk [ here a,b,c,p,q,r are constant values ]

<3,1,-2>= <a,b,c> + <p,q,r>

<3,1,-2> = <a+p , b+q, c+r>

comparing a+p=3 , b+q=1 , c+r=-2

given u1 is parallel to v

then u1xv =0

<a,b,c> x <3,-1,-3> =0

<-3b+c ,3a+3c , -a-3b >=0

-3b+c=0 a+c =0 a+3b=0

c=3b a =-3b

u1= <a,b,c> = <-3b , b , 3b> = b<-3,1,3>

u2 is perpendicular to v

u2.v=0

<p,q,r>.<3,-1,-3>=0

3p -q-3r=0

3p= q+3r

now p =6 ,q=0 r=-5

u1=<-3,1,3> and u2=<6,0,5>