To determine if the points with the given position vectors are collinear, we need to set up vectors
→
AB
and
→
AC
as follows: Let point A have the position vector α‌hat‌i+10‌hat‌j+13‌hat‌k. Let point B have the position vector 6‌hat‌i+11‌hat‌j+11‌hat‌k. Let point C have the position vector ‌
9
2
‌hat‌i+β‌hat‌j−8‌hat‌k. The vector
→
AB
is calculated as:
→
AB
=(6−α)‌hat‌i+(11−10)‌hat‌j+(11−13)‌hat‌k So:
→
AB
=(6−α)‌hat‌i+hat‌j−2‌hat‌k The vector
→
AC
is calculated as:
→
AC
=(‌
9
2
−α)‌hat‌i+(β−10)‌hat‌j+(−8−13)‌hat‌k So:
→
AC
=(‌
9
2
−α)‌hat‌i+(β−10)‌hat‌j−21‌hat‌k Since the points are collinear, vectors
→
AB
and
→
AC
are parallel, which implies: ‌
6−α
9
2
−α
=‌
1
β−10
=‌
−2
−21
Solving these equations, we find: ‌‌
1
β−10
=‌
2
21
⇒β−10=‌
21
2
‌β=10+‌
21
2
=‌
41
2
For the other ratio: ‌‌
12−2α
9−2α
=‌
2
21
‌126−21α=9−2α So: 19α=117 α=‌
117
19
=6 Given: (19α−6β)2=(117−123)2=(−6)2=36 Thus, the final result is (19α−6β)2=36.