Solving for Orthogonal Vectors in R4?

[concon]

Homework Statement

Given following vectors in R4:
v= (4,-9,-6,3)
u = (5,-8,k,4)
w=(s,-5,4,t)

A. Find value of k if u and v are orthogonal
B. Find values of S and T if w and u are orthogonal and w and v are orthogonal

Homework Equations

Orthogonal means dot product is zero

The Attempt at a Solution

A. I already solved and got k= 104/6 which is correct


B.
I tried to solve this one assuming that k was still equal to 104/6 and got
s=4 and t=-37/3
But this is not the answer.
Do I assume that k is still the same result as in part A or can k be anything?
If k is anything than I get:

[
4 3 -21
5 4 (-40-4k)
]
as a matrix, but how would I solve this?

 

[jbunniii]

I think you should continue to use $k=104/6$ in part B. Can you show how you calculated $s$ and $t$? You can easily check that with those values, $w$ is orthogonal to $v$ but not to $u$.

 

[concon]

I think you should continue to use $k=104/6$ in part B. Can you show how you calculated $s$ and $t$? You can easily check that with those values, $w$ is orthogonal to $v$ but not to $u$.

Okay that's what I thought b/c if k is unknown you have 3 variables which wouldn't be solveable.

So if k= 104/6
v*w = 4s + 45 - 24 + 3t = 0
u*w = 5s + 40 + 416/6 + 4t = 0

From there I combined like terms:

4s + 3t = -21
5s + 4t = -656/6

Then I formed a matrix and use row operations to find that:

s = 244
t = -997/3

Is this right? I think my mistake the first time using this method was when I got the 40 and 416/6 on the right hand side I added 40 instead of subtracting which changed the value.

 

[jbunniii]

Yes, that looks right. You can always substitute your answers back into your expressions for v*w and u*w to verify that you get zero in both cases.