How can I find the vector from one point to another using their coordinates?
Quick answer
Subtract the starting point’s coordinates from the ending point’s coordinates, in that order. For points A = (1, 2) and B = (4, 6), the vector from A to B is (4 − 1, 6 − 2) = (3, 4). The components show how each coordinate changes. Reversing the points gives (−3, −4), the opposite displacement. [S1]
The question, the details, the next step.
What’s the rule for finding the vector from one point to another? I keep mixing up which point to subtract.
Name the points by their roles first: let A be the starting point and B the ending point. Then subtract A’s coordinates from B’s coordinates, coordinate by coordinate. If A = (xₐ, yₐ) and B = (xᵦ, yᵦ), the displacement vector from A to B is (xᵦ − xₐ, yᵦ − yₐ). In other words, use ending minus starting, not the reverse. [S1]
A useful way to remember the order is to say the route aloud: “from A to B.” The vector records the change needed to move from the first point to the second. Keep the coordinate order consistent too: subtract x-values to get the first component, and y-values to get the second.
Can you show the calculation with small numbers and explain what the result tells me?
Suppose the starting point is A = (1, 2) and the ending point is B = (4, 6). Subtract the starting coordinates from the ending coordinates:
- x-component: 4 − 1 = 3.
- y-component: 6 − 2 = 4.
- So the vector from A to B is (3, 4). [S1]
The first component says the x-coordinate increases by 3; the second says the y-coordinate increases by 4. The pair describes the displacement, not the ending point itself. Check by adding the vector to the start: (1 + 3, 2 + 4) = (4, 6), which recovers B.
What if one of the coordinates goes down? Does a negative component mean I did the subtraction wrong?
Not necessarily. A negative component means that coordinate decreased from start to finish. For example, if the start is (5, 2) and the end is (2, 5), then the vector is (2 − 5, 5 − 2) = (−3, 3). The x-coordinate went down by 3, while the y-coordinate went up by 3. [S1]
To check your direction, compare each component with the corresponding coordinate change in the points. If the ending x-value is smaller than the starting x-value, the x-component should be negative. If it is larger, the x-component should be positive. Apply the same check separately to y. This helps catch a reversed subtraction without assuming that every component must be positive.
How can I check the direction by reversing the points?
Use the same two points but swap their roles. For A = (1, 2) and B = (4, 6), the vector from A to B is (3, 4). From B to A, subtract B from A instead: (1 − 4, 2 − 6) = (−3, −4). Both components have changed sign, so the reversed displacement points in the opposite direction. [S1]
For a final direction check, add your proposed vector to the stated start point. It should land on the stated end point. Then swap the points: the reverse vector should undo the first change. [S1]
A practical checklist
Label which point is the start and which is the end.
Subtract start x from end x, then start y from end y.
Keep negative components when an ending coordinate is smaller.
Add the vector to the starting point and check that you reach the endpoint.
For the reverse direction, swap the points and check that both component signs reverse.
A few more useful answers
Does the order of the coordinates matter?
Yes. Keep the coordinate order consistent: the first component comes from the x-coordinates and the second from the y-coordinates. Also keep the point order consistent: ending minus starting. [S1]
What should I do if the two points are identical?
Subtract the coordinates in the usual order. Each difference is zero, so the displacement vector is (0, 0). [S1]
Does this guide establish how to find a three-dimensional vector?
The cited explanation and examples establish the rule for two-dimensional Cartesian coordinates. They do not provide a three-dimensional example, so follow the coordinate conventions given in your course or problem. [S1]
Sources
References used for this guide.
- [S1] OpenStax University Physics Volume 1: Coordinate Systems and Components of a Vector openstax.org · Checked