Show that vectors a b c are coplanar iff a+b b+c c+a are coplanar
Answers
Answered by
2
Answer:
Step-by-step explanation:
Vectors a, b and c are co-planar if
[a b c] = 0
Finding [a + b b + c c + a]
[a + b b + c c + a]
= (a + b).[(b + c) x (c + a)]
= (a + b).[(b x c) + (b x a) + (c x c) + (c x d)], c x c = 0,
=a.(b x c) + b.(b x c) + a.(b x a) + b.(b x a) + a.(c x a) + b.(c x a)
= [a, b, c] + [b, b, c] + [a, b, a] + [b, b, a] + [a, c, a] + [b, c, a]
We have that [b, b, c] = [c, b, b] which is equal to 0 because b x b = 0.
Then we apply it to the following too:
[a, b, a] = 0
[b, b, a] = 0
[a, c, a] = 0
Then:
= [a, b, c] + 0 + 0 + 0 + 0 + [b, c, a]
As [a, b, c] = [c, a, b] = [b, c, a]
= [a, b, c] + [a, b, c]
= 2[a, b, c]
= 2 x 0
= 0
Since [a+ b b + c c + a] = 0
a+b b+c and c+a are also co-planar.
Hence proved.
Similar questions