Tutorial questions - Algebra of vectors D 11. Suppose \( A B C \) is the triangle with vertices \( A(1,1), B(4,3) \), and \( C(6,0) \). (a) Find the vectors \( A B, B C \), and \( C A \) and show that their sum is \( \mathbf{0} \) (the zero vect Use a diagram to explain why this is so.

To find the vectors \( A B, B C \), and \( C A \), you will first compute the components of each vector. 1. \( A B = B - A = (4, 3) - (1, 1) = (3, 2) \) 2. \( B C = C - B = (6, 0) - (4, 3) = (2, -3) \) 3. \( C A = A - C = (1, 1) - (6, 0) = (-5, 1) \) Now, adding these vectors together: \[ A B + B C + C A = (3, 2) + (2, -3) + (-5, 1) = (0, 0) \] So their sum is indeed \( \mathbf{0} \). In a diagram, this relationship can be visualized by placing the triangle \( ABC \) in a coordinate grid. If you draw arrows for each vector, starting from the tail of one vector to the head of the next (like \( A B \) leading into \( B C \), and so on), you'll see that they form a closed loop, returning to the starting point, which confirms that the sum equals the zero vector \((0, 0)\). This closure around the triangle is visually compelling, illustrating that the combined movement along the edges of the triangle returns you to your original point.