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:

\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:

\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}
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