If ` vec xa n d vec y` are two non-collinear vectors and a, b, and c represent the sides of a ` A B C` satisfying `(a-b) vec x+(b-c) vec y+(c-a)( vec x xx vec y)=0,` then ` A B C` is (where ` vec xxx vec y` is perpendicular to the plane of `xa n dy` ) a. an acute-angled triangle b. an obtuse-angled triangle c. a right-angled triangle d. a scalene triangle

Explanation

To solve the problem, we need to analyze the given equation involving the sides of triangle ABC and the vectors \(\vec{x}\) and \(\vec{y}\).

### Step-by-Step Solution:

1. **Understand the Given Equation**:
The equation given is:
\[
(a-b) \vec{x} + (b-c) \vec{y} + (c-a)(\vec{x} \times \vec{y}) = 0
...