Given the following vectors in R^4 R 4 : v=(-6,-1,-7,-7), u=(-5,3,k,-6) w=(s,-2,

Question

Given the following vectors in R^4R 4 :
v=(-6,-1,-7,-7),
u=(-5,3,k,-6)
w=(s,-2,-4,t) v=(?6,?1,?7,?7), u=(?5,3,k,?6), w=(s,?2,?4,t) (a) Find the value kk such that the vector u u is orthogonal to the vector vv , and then
(b) Find the value ss and tt such that the vector w w is orthogonal to the both vectors vv and uu .
Given the following vectors in R^4R 4 :
v=(-6,-1,-7,-7),
u=(-5,3,k,-6)
w=(s,-2,-4,t) v=(?6,?1,?7,?7), u=(?5,3,k,?6), w=(s,?2,?4,t) v=(?6,?1,?7,?7), u=(?5,3,k,?6), w=(s,?2,?4,t) (a) Find the value kk such that the vector u u is orthogonal to the vector vv , and then
(b) Find the value ss and tt such that the vector w w is orthogonal to the both vectors vv and uu .

Explanation / Answer

I am assuming according to your question

a)

u is orthogonal to the vector v

So u.v=0

Given v=(?6,?1,?7,?7), u=(?5,3,k,?6)

u.v = -6*-5+3*-1+k*-7+(-6*-7) = 0

30-3-7k+42=0

k=69/7

b)

Also given

w is orthogonal to  v

w is orthogonal to  u

w.v=0

w.u=0

v=(?6,?1,?7,?7), u=(?5,3,k,?6), w=(s,?2,?4,t)

w.v=0

-6s+2+28-7t=0 ----->1

w.u=0

-5s-6-4k-6t =0 ---------->2

We know k=69/7

equation 2 becomes

-5s-6-276/7-6t =0 ---------->2

we have

6s+7t=30

5s+6t = -318/7

solving 1 and 2 we have

s= 498 ,t = -2958/7

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