Let a,b,c denote the lengths of the sides of a triangle such that
` ( a-b) vecu + ( b-c) vecv + ( c-a) (vecu xx vecv) = vec0`
For any two non-collinear vectors ` vecu and vecu`,then the triangle is

Options

A. right angled
B. equilateral
C. isoscels
D. obtuse angled

Explanation

To solve the problem, we need to analyze the given vector equation and its implications on the triangle formed by the sides \( a, b, c \).

### Step-by-step Solution:

1. **Understand the Given Equation**:
The equation given is:
\[ (a - b) \vec{u} + (b - c) \vec{v} + (c - a)(\vec{u} \times \vec{v}) = \vec{0} \]
This equation involves three terms, each multiplied by a vector.

2. **Recognize the Implications of Non-Collinearity**:
Since \( \vec{u} \) and \( \vec{v} \) are non-collinear vectors, it implies that \( \vec{u} \), \( \vec{v} \), and \( \vec{u} \times \vec{v} \) are non-coplanar. This means they do not lie in the same plane.

3. **Analyze the Vector Equation**:
For the sum of these three vectors to equal the zero vector, and since they are non-coplanar, each coefficient of the vectors must individually equal zero. Thus, we can set up the following equations:
\[ a - b = 0 \]
\[ b - c = 0 \]
\[ c - a = 0 \]

4. **Solve the Equations**:
From \( a - b = 0 \), we get:
\[ a = b \]
From \( b - c = 0 \), we get:
\[ b = c \]
From \( c - a = 0 \), we get:
\[ c = a \]

5. **Conclude the Relationship Between Sides**:
Since \( a = b \), \( b = c \), and \( c = a \), we conclude that:
\[ a = b = c \]
This means all sides of the triangle are equal.

6. **Identify the Type of Triangle**:
A triangle with all sides equal is known as an equilateral triangle.

### Final Conclusion:
The triangle is equilateral.