Question

How do I write vectors whose initial point is not at the origin?

Answer 1

It is sometimes necessary to consider vectors whose initial points are not at the origin. If $\overrightarrow{P_{1} P_{2}}$ denotes the vector with initial point $P_{1}\left(x_{1}, y_{1}\right)$ and terminal point $P_{2}\left(x_{2}, y_{2}\right)$, then the components of this vector are given by the formula
$$
\overrightarrow{P_{1} P_{2}}=\left(x_{2}-x_{1}, y_{2}-y_{1}\right)
$$
That is, the components of $\overrightarrow{P_{1} P_{2}}$ are obtained by subtracting the coordinates of the initial point from the coordinates of the terminal point. For example, in Figure 3.1.12 the vector $\overrightarrow{P_{1} P_{2}}$ is the difference of vectors $\overrightarrow{O P_{2}}$ and $\overrightarrow{O P_{1}}$, so
$$
\overrightarrow{P_{1} P_{2}}=\overrightarrow{O P_{2}}-\overrightarrow{O P_{1}}=\left(x_{2}, y_{2}\right)-\left(x_{1}, y_{1}\right)=\left(x_{2}-x_{1}, y_{2}-y_{1}\right)
$$
As you might expect, the components of a vector in 3 -space that has initial point $P_{1}\left(x_{1}, y_{1}, z_{1}\right)$ and terminal point $P_{2}\left(x_{2}, y_{2}, z_{2}\right)$ are given by
$$
\overrightarrow{P_{1} P_{2}}=\left(x_{2}-x_{1}, y_{2}-y_{1}, z_{2}-z_{1}\right)
$$