Question

Show that vectors a b c are coplanar iff a+b b+c c+a are coplanar

Answer 1

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.