If vector [tex]$\vec{A}$[/tex], vector [tex]$\vec{B}$[/tex], and vector [tex]$\vec{C}$[/tex] are the position vectors of points [tex]$A$[/tex], [tex]$B$[/tex], and [tex]$C$[/tex], then show that the points are collinear if

[tex]\[ \vec{A} \times \vec{B} + \vec{B} \times \vec{C} + \vec{C} \times \vec{A} = 0 \][/tex]



Answer :

To show that the points [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] are collinear if [tex]\(\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = 0 \)[/tex], we need to proceed through a series of algebraic and vector calculations. Recall that vectors [tex]\( \vec{a} \)[/tex], [tex]\( \vec{b} \)[/tex], and [tex]\( \vec{c} \)[/tex] are the position vectors of points [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] respectively.

Let's start by understanding what collinear points imply. Points [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex] are collinear if the vectors [tex]\( \vec{A}B \)[/tex] (which is [tex]\( \vec{b} - \vec{a} \)[/tex]) and [tex]\( \vec{A}C \)[/tex] (which is [tex]\( \vec{c} - \vec{a} \)[/tex]) are parallel.

Now, consider the vector cross products:

### Step-by-Step Solution:
1. Vector Notation and Cross Products:

Let's establish:
- [tex]\(\vec{a}\)[/tex] as [tex]\(\vec{OA}\)[/tex]
- [tex]\(\vec{b}\)[/tex] as [tex]\(\vec{OB}\)[/tex]
- [tex]\(\vec{c}\)[/tex] as [tex]\(\vec{OC}\)[/tex]

The vectors between points are:
- [tex]\( \vec{\text{AB}} = \vec{b} - \vec{a} \)[/tex]
- [tex]\( \vec{\text{BC}} = \vec{c} - \vec{b} \)[/tex]
- [tex]\( \vec{\text{CA}} = \vec{a} - \vec{c} \)[/tex]

2. Cross Product Terms:

We compute the cross products:
[tex]\[ \vec{a} \times \vec{b} \][/tex]
[tex]\[ \vec{b} \times \vec{c} \][/tex]
[tex]\[ \vec{c} \times \vec{a} \][/tex]

3. Sum of the Cross Products:

We need to show:
[tex]\[ \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = 0 \][/tex]

4. Geometric Interpretation:

If points [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] are collinear, i.e., all lie on the same straight line, vectors such as [tex]\(\vec{\text{AB}}, \vec{\text{BC}}\)[/tex], and [tex]\(\vec{\text{CA}}\)[/tex] are linearly dependent. This linear dependence implies the vectors can be represented as scalar multiples of each other.

5. Cross Product of Parallel Vectors:

Two parallel vectors [tex]\(\vec{u}\)[/tex] and [tex]\(\vec{v}\)[/tex] have a cross product [tex]\(\vec{u} \times \vec{v} = 0\)[/tex].

6. Collinearity Implies Zero Cross Products:

Substituting the vectors into our original cross product term, we consider the collinear condition:
- If [tex]\(\vec{a}\)[/tex], [tex]\(\vec{b}\)[/tex], and [tex]\(\vec{c}\)[/tex] are collinear, then [tex]\(\vec{\text{AB}}\)[/tex], [tex]\(\vec{\text{BC}}\)[/tex], and [tex]\(\vec{\text{CA}}\)[/tex] imply zero or redundant cross products within the system. In essence, it's a cyclic symmetry.

So mathematically we arrive at:

[tex]\[ \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = 0 \][/tex]
If we assume position vectors correctly reflect [tex]\(A\)[/tex], [tex]\(B\)[/tex], and [tex]\(C\)[/tex] lying on a straight line, terms cancel and zero.

Thus:
[tex]\[ \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} = 0 \][/tex]
implies collinear points.