Q2) Solve the following questions. (a) Find the terminal point of the vector that is equivalent to \( \mathbf{u}=(1,2) \) and whose initial point is \( A(1,1) \). (b) Find the initial point of the vector that is equivalent to \( \mathbf{u}=(1,1,3) \) and whose terminal point is \( B(-1,-1,2) \).

To solve the problems, we will use the properties of vectors and their representation in a coordinate system. ### (a) Finding the terminal point of the vector \( \mathbf{u} = (1, 2) \) with initial point \( A(1, 1) \) The terminal point \( T \) of a vector \( \mathbf{u} \) with initial point \( A(x_1, y_1) \) can be found using the formula: \[ T = A + \mathbf{u} \] Given: - \( A(1, 1) \) - \( \mathbf{u} = (1, 2) \) We can express this as: \[ T = (1, 1) + (1, 2) = (1 + 1, 1 + 2) = (2, 3) \] Thus, the terminal point is: \[ \boxed{(2, 3)} \] ### (b) Finding the initial point of the vector \( \mathbf{u} = (1, 1, 3) \) with terminal point \( B(-1, -1, 2) \) To find the initial point \( A \) of the vector \( \mathbf{u} \) with terminal point \( B(x_2, y_2, y_3) \), we can rearrange the formula: \[ A = B - \mathbf{u} \] Given: - \( B(-1, -1, 2) \) - \( \mathbf{u} = (1, 1, 3) \) We can express this as: \[ A = (-1, -1, 2) - (1, 1, 3) = (-1 - 1, -1 - 1, 2 - 3) = (-2, -2, -1) \] Thus, the initial point is: \[ \boxed{(-2, -2, -1)} \]