## Video Lesson: How to Find the Vector Between Two Points

## How to Find the Vector Between Two Points

**To find the vector between two points, subtract the coordinates of the initial point from the coordinates of the final point. **

**For example, the vector from A(2, 3) to B(4, 0) is found by subtracting A from B with the calculation (4-2, 0-3) to obtain (2, -3).**

It is important to note the order of the subtraction.

In the calculation of any vector , we subtract the coordinates of A from the coordinates of B. Subtract the first set of coordinates listed from the second set of coordinates.

Point A has the coordinates (2, 3) and point B has the coordinates (4, 0).

The vector of is simply the vector from the origin to point A. It contains the same values as the coordinates of point A.

Therefore the vector and .

The vector is the difference between A and B. It describes how to travel from point A to point B.

We calculate it using the formula: .

When we subtract vectors, we subtract the ** i** components from each other and the

**components from each other separately.**

*j*That is, we subtract the numbers on the tops of the vector and then the numbers on the bottom of the vector.

Therefore which is evaluated to be .

We can see the vector from A to B in the diagram above shown with the red arrow. We can see from the diagram that this matches the calculated vector because it travels 2 right and 3 down.

Here is another example of calculating a vector between two points.

Given the points P(4, -1) and Q(-3, 0), find the vector .

Since the vector is directed from P to Q, we subtract P from Q.

We subtract to obtain .

Evaluating this, .

The vector from A to B is equal to the distance of each component from point A to point B.

An alternative method to calculate the vector from A to B is to work out the values that must be added to point A to get to point B.

We think ‘what must be added to the first set of coordinates to obtain the second set of coordinates’.

In the above example, we need to get from to .

We must subtract 7 from 4 to obtain -3 and we must add 1 to -1 to obtain 0.

Therefore the vector .

## How to Find the Vector Between Two Points in 3D

**To find a vector between two points in 3D, subtract the initial point from the final point. The vector from point A(a _{1}, a_{2}, a_{3}) to point B(b_{1}, b_{2}, b_{3}) is given by (b_{1} – a_{1}, b_{2} – a_{2} , b_{3} – a_{3}).**

Simply subtract the x coordinates of point A from the x coordinates of point B, the y coordinates of point A from the y coordinates of point B and the z coordinates of point A from the z coordinates of point B

For example, find the vector from A(3, 1, 2) to B(-2, 0, 3).

- For the
component of the vector: -2 – 3 = -5*i* - For the
component of the vector: 0 – 1 = -1*j* - For the
**k**component of the vector: 3 – 2 = 1

The vector 3D vector from A to B is .

Again, we can also calculate this by thinking ‘what vector must be added to the first point of to obtain the second point of ?’

We know that and so the vector .

## Vector Between Two Points Formula

**The formula to calculate a vector from A to B is (A→B) = (O→B) – (O→A). In words this means to subtract the coordinates of point B from point A.**

The formula for the vector from A to B

The vector is simply made up of the coordinates of point A, which are and similarly, the vector is .

Therefore, the formula for the vector can be written in 2 dimensions as:

The formula for the vector from A to B in 2D

In 3 dimensions, the vector

The formula for the vector from A to B in 3D

## Examples of Calculating the Vector Between Two Points

Here are some examples of calculating the vector between two points A and B in 2 dimensions.

The ** i** component of the vector is found by subtracting the x coordinate of A from the x coordinate of B.

The ** j** component of the vector is found by subtracting the y coordinate of A from the y coordinate of B.

Point A | Point B | i component calculation | j component calculation | Vector from A to B |

(0,0) | (4, 3) | 4 – 0 = 4 | 3 – 0 = 3 | (4, 3) |

(1, 2) | (8, 3) | 8 – 1 = 7 | 3 – 2 = 1 | (7, 1) |

(6, 3) | (3, 1) | 3 – 6 = -3 | 1 – 3 = -2 | (-3, -2) |

(-5, -2) | (3, 0) | 3 – -5 = 8 | 0 – -2 = 2 | (8, 2) |

(-3, 4) | (-1, -2) | -1 – -3 = 2 | -2 – 4 = -6 | (2, -6) |

Here are some examples of calculating the vector between two points A and B in 3 dimensions.

The ** i** component of the vector is found by subtracting the x coordinate of A from the x coordinate of B.

The ** j** component of the vector is found by subtracting the y coordinate of A from the y coordinate of B.

Point A | Point B | i component calculation | j component calculation | k component calculation | Vector from A to B |

(0, 0, 0) | (1, 3, 2) | 1 – 0 = 1 | 3 – 0 = 3 | 2 – 0 = 2 | (1, 3, 2) |

(1, 5, 3) | (4, 8, 6) | 4 – 1 = 3 | 8 – 5 = 3 | 6 – 3 = 3 | (3, 3, 3) |

(8, 4, 1) | (2, 3, 1) | 2 – 8 = -6 | 3 – 4 = -1 | 1 – 1 = 0 | (-6, -1, 0) |

(-1, 0, -2) | (3, 2, 1) | 3 – -1 = 4 | 2 – 0 = 2 | 1 – -2 = 3 | (4, 2, 3) |

(-5, -1, -3) | (-2, -4, -9) | -2 – -5 = 3 | -4 – -1 = -3 | -9 – -3 = -6 | (3, -3, -6) |

## Vector Between Two Points Calculator

Below is a calculator to find the vector between two points.

Select whether the vector is in 2 or 3 dimensions and then enter the x, y and z coordinates of each below.

The calculator will then calculate the i, j and k components of the vector from A to B.