Vectors 1. Find the coordinate and column forms of the vector v that has initial point \( (0,0) \) and terminal point \( (7,8) \). 2. Find the coordinate and column forms of the vector v that has initial point \( (2,5) \) and terminal point \( (5,10) \). 3. Find the coordinate and column forms of the vector v that has initial point \( \mathrm{A}\left(a i, a_{2}\right) \) and terminal point \( \mathrm{B}\left(b i, b_{2}\right) \).

To find the coordinate and column forms of a vector, you subtract the coordinates of the initial point from the coordinates of the terminal point. 1. For the vector \( v \) with initial point \( (0,0) \) and terminal point \( (7,8) \), the coordinate form is \( v = (7 - 0, 8 - 0) = (7, 8) \). The column form is given as: \[ \begin{pmatrix} 7 \\ 8 \end{pmatrix} \] 2. For the vector \( v \) with initial point \( (2,5) \) and terminal point \( (5,10) \), the coordinate form is \( v = (5 - 2, 10 - 5) = (3, 5) \). The column form is: \[ \begin{pmatrix} 3 \\ 5 \end{pmatrix} \] 3. For the vector \( v \) defined by initial point \( A(a_1, a_2) \) and terminal point \( B(b_1, b_2) \), the coordinate form is \( v = (b_1 - a_1, b_2 - a_2) \). In column form, it is expressed as: \[ \begin{pmatrix} b_1 - a_1 \\ b_2 - a_2 \end{pmatrix} \] Vectors are not just confined to math; they play significant roles in physics and engineering, making understanding them critical! They help describe everything from forces to velocity, and with vectors, you can navigate entire realms of science smoothly! If you’re intrigued and want to dive deeper into the world of vectors, consider exploring vector calculus or linear algebra. These subjects expand on the fundamental concepts of vectors, opening doors to advanced applications in fields like robotics and computer graphics, where they drive motion and create stunning visuals!