# What is meant by the initial point of a vector?

Aug 21, 2015

Geometrically, a vector is a length in a direction.

#### Explanation:

A vector is (or can be thought of as) a directed line segment.

A vector (unlike a line segment) goes from one point to another.

A line segment has two endpoints and a length. It is a length in a particular location.

A vector has only a length and a direction. But we like to represent vectors using line segments.

When we try to represent a vector using a line segment, we need to distinguish one direction along the segment from the other direction. Part of doing this (or one way of doing it) is to distinguish the two endpoints by labeling one of them "initial" and the other "terminal"

For example, using 2 dimensional coordinates:

There is a line segment connecting the points $\left(0 , 1\right)$ and $\left(5 , 1\right)$. We can describe the same segment by saying that it connects $\left(5 , 1\right)$ and $\left(0 , 1\right)$. (It is a horizontal line segment of length $5$.)

There as also a vector from $\left(0 , 1\right)$ to $\left(5 , 1\right)$. (Some ways of describing it: the x coordinates are increasing, the vector points to the right, the initial point is $\left(0 , 1\right)$, the terminal point is $\left(5 , 1\right)$.)

and a different vector from $\left(5 , 1\right)$ to $\left(0 , 1\right)$ (The x ccordinates are decreasing, the vector points to the left, the initial point is $\left(5 , 1\right)$, the terminal point is $\left(0 , 1\right)$.)

The vector from $\left(4 , 7\right)$ to $\left(9 , 7\right)$ is the same vector as from $\left(0 , 1\right)$ to $\left(5 , 1\right)$, (It has the same magnitude and the same direction.)
But it has a different initial point.