Determining a Vector Given Two Points

Remember that a vector consists of both an initial point and a terminal point. Because of this, we can write vectors in terms of two points in certain situations.

# Vectors with Initial Points at The Origin

Let's say we have two points in 3-space, one of which has its initial point situated at the origin $O(0, 0, 0)$ and its terminal point at coordinates $P(3, 2, 1)$. We can draw the vector $\vec{OP}$ as follows:

In these cases, the direction of the arrow on top of vector $\vec{OP}$ corresponds to the initial and terminal points of the vector $\vec{OP}$, that is the arrow indicates the vector goes from point $O$ to point $P$. Thus in general, $\vec{OP} \neq \vec{PO}$ since $\vec{PO}$ has its initial point at $P$ and terminal point at the origin. The only difference between these vectors in their direction, and hence we can see that $\vec{OP} = -\vec{PO}$.

# Vectors with Initial Points NOT at The Origin

Sometimes we don't place the initial point of a vector at the origin. For example, consider a vector that has its initial point at $P(2, 2)$ and terminal point at $Q(6, 3)$. To draw this vector, we can plot these coordinates and connect them as a vector. Alternatively we can denote this vector with a general set of components:

(1)
\begin{align} \vec{PQ} = (x_{Q} - x_{P}, y_{Q} - y_{P}) \end{align}

For our example, $\vec{PQ} = (4, 1)$, and the following graphic illustrates our vector in two ways:

## Example 1

Given that a vector $\vec{PQ}$ has an initial point at $P(2, 2, 1)$ and a terminal point at $Q(6, 3, 2)$, find the vector $\vec{PQ}$:

To do this, we will simply subtract point $P$ from point $Q$ to obtain:

(2)
\begin{align} \vec{PQ} = (x_Q - x_P, y_Q - y_P, z_Q - z_P) \\ \vec{PQ} = (6 - 2, 3 - 2, 2 - 1) \\ \vec{PQ} = (4, 1, 1) \end{align}