Question

Show that the vectors a b c are coplanar if and only if a + b, b + c, c plus are coplanar

Answer 1

Answer:

The proof is given below.

Step-by-step explanation:

Given  the vectors a b c . we have to prove that if a b c are coplanar then $\vec a.(\vec b\times \vec c)$

As we know, if the three vectors are coplanar then their scalar triple product equals to 0.

Therefore, $[\vec a\thinspace \thinspace \vec b \thinspace \thinspace \vec c]=0$

Now, $[\vec a+\vec b\thinspace \thinspace \vec b+\vec c \thinspace \thinspace \vec c+\vec a]$

⇔ $(\vec a+\vec b).[(\vec b+\vec c)\times (\vec c+\vec a)$

⇔ $(\vec a+\vec b).[(\vec b\times \vec c)+(\vec b\times \vec a)+(\vec c\times \vec c)+(\vec b\times \vec a)$

⇔ $(\vec a+\vec b).[(\vec b\times \vec c)+(\vec b\times \vec a)+0+(\vec b\times \vec a)$

⇔$\vec a.(\vec b\times \vec c)+\vec b.(\vec b\times \vec c)+\vec a.(\vec b\times \vec a)+\vec b.(\vec b\times \vec a)+\vec a.(\vec c\times \vec a)+\vec b.(\vec c\times \vec a)$

⇔$2(\vec a.(\vec b\times \vec c))=0$

Hence, a + b, b + c, c plus are coplanar.

By reversing steps we can prove that if a + b, b + c, c plus are coplanar then a, b , c are coplanar.

Hence, a b c are coplanar if and only if a + b, b + c, c plus are coplanar

Answer 2

Answer:

The three vectors are coplanar if their scalar triple product is zero.

i.e. a,b and c are coplanar that means:

a·(b×c)=0.  ( This expression is same as: [a,b,c]  )

Now:

[a+b,b+c,c+a]=(a+b)·[(b+c)×(c+a)]

                     =(a+b)·[b×c+b×a+c×c+c×a]

                    = (a+b)··[b×c+b×a+0+c×a]

(since cross product of same vector is zero.

as c×c=|c|.|c| sin θ

and θ=0 for same vectors.

so, sin θ=0 )

Hence,

[a+b,b+c,c+a]=(a+b).(b×c+b×a+c×a)

            =a·(b×c)+a·(b×a)+a·(c×a)+b·(b×c)+b·(b×a)+b·(c×a)

            =[a,b,c]+[a,b,a]+[a,c,a]+[b,b,c]+[b,b,a]+[b,c,a]

Also,

[a,b,a]=[a,c,a]=[b,b,c]=[b,b,a]=0

⇒    [a+b,b+c,c+a]=[a,b,c]+[b,c,a]

⇒   [a+b,b+c,c+a]=2[a,b,c]

Hence, [a+b,b+c,c+a]=0 if and only if [a,b,c]=0.

Hence, we can say that:

the vectors a,b,c are coplanar if and only if a + b, b + c, c plus are coplanar.