Let ` veca , vecb , vecc` represent respectively `vec(BC), vec(CA) and vec(AB) ` where ABC is a triangle , Then ,

Options

A. `veca + vecb = vecc`
B. ` vecb + vecc = veca`
C. ` veca xx vecb = vecb xx vecc = vecc xx veca`
D. ` [ veca vecb vecc] = [ vecb vecc veca] = [ vecc veca vecb] ne 0`

Explanation

To solve the problem, we need to analyze the relationships between the vectors representing the sides of triangle ABC. Let's denote the vectors as follows:

- \( \vec{a} = \vec{BC} \)
- \( \vec{b} = \vec{CA} \)
- \( \vec{c} = \vec{AB} \)

### Step 1: Understand the relationships between the vectors
In triangle ABC, the vectors can be expressed in terms of each other. By the triangle law of vector addition, we can write:

\[ \vec{a} + \vec{b} + \vec{c} = 0 \]

This means that the sum of the vectors representing the sides of the triangle is zero.

### Step 2: Cross product of the vectors
Now, we will take the cross product of the entire equation with one of the vectors. Let's take the cross product with \( \vec{a} \):

\[ \vec{a} \times (\vec{a} + \vec{b} + \vec{c}) = \vec{a} \times 0 \]

Since the cross product of any vector with itself is zero, we have:

\[ \vec{a} \times \vec{a} + \vec{a} \times \vec{b} + \vec{a} \times \vec{c} = 0 \]

This simplifies to:

\[ \vec{0} + \vec{a} \times \vec{b} + \vec{a} \times \vec{c} = 0 \]

Thus, we can conclude:

\[ \vec{a} \times \vec{b} + \vec{a} \times \vec{c} = 0 \]

### Step 3: Rearranging the equation
From the previous step, we can rearrange the equation:

\[ \vec{a} \times \vec{b} = -\vec{a} \times \vec{c} \]

### Step 4: Similar calculations for other pairs
We can similarly take the cross product with \( \vec{b} \) and \( \vec{c} \):

1. Taking the cross product with \( \vec{b} \):
\[ \vec{b} \times (\vec{a} + \vec{b} + \vec{c}) = 0 \implies \vec{b} \times \vec{a} + \vec{b} \times \vec{b} + \vec{b} \times \vec{c} = 0 \]
This simplifies to:
\[ \vec{b} \times \vec{a} + \vec{b} \times \vec{c} = 0 \implies \vec{b} \times \vec{a} = -\vec{b} \times \vec{c} \]

2. Taking the cross product with \( \vec{c} \):
\[ \vec{c} \times (\vec{a} + \vec{b} + \vec{c}) = 0 \implies \vec{c} \times \vec{a} + \vec{c} \times \vec{b} + \vec{c} \times \vec{c} = 0 \]
This simplifies to:
\[ \vec{c} \times \vec{a} + \vec{c} \times \vec{b} = 0 \implies \vec{c} \times \vec{a} = -\vec{c} \times \vec{b} \]

### Step 5: Conclusion
From the above steps, we have established that:

\[ \vec{a} \times \vec{b} = \vec{b} \times \vec{c} = \vec{c} \times \vec{a} \]

Thus, the correct answer is that all these cross products are equal.