Question

If possible, find a, b, and c so that
v =
[a
b
c]
is orthogonal to both
w =
[1
6
1]
and x =
[1
−9
1]

Answer 1

A vector that is orthogonal to two given vectors is the cross product, namely

|<b>i j k</b>|
|x1 y1 z1|
|x2 y2 z2|
which in this case is
|<b>i j k</b>|
|1 6 1|
|1 -9 1|

Expanding the determinant, we get the following vector:
<6*1+1*9, 1*1-1*1, 1*-9-6*1>
<15, 0, -15>
which can be reduced to
<1,0,-1>
Check:
<1,6,1>. <1,0,-1>=6-0-6=0
<1,-9,1>. <1,0,-1>=1+0-1=0
Thus <1,0,-1> is orthogonal to both given vectors.
<15,0

Answer 2

how did you expand the determiannt

Answer 3

thank you